Cluster Extensions of Dynamical Mean-Field Theory · For a small cluster and likewise for a system of disconnected clusters, even for the interacting case, it is comparatively simple

Autumn School on Correlated Electrons:DMFT - From Infinite Dimensions to Real Materials

Cluster Extensions !of Dynamical Mean-Field Theory

Michael Potthoff !I. Institute of Theoretical Physics, Hamburg University!

outline:!• motivation!• geometry!• Green’s functions!• cluster-perturbation theory!• periodization schemes!• self-consistent cluster embedding approaches!• applications

(Sep. 17-21, 2018, FZ Jülich)

MOTIVATION

timeline

1966: Hubbard-I approximation ! |!1993: cluster-perturbation theory (CPT)! |!1989 - 1992: dynamical mean-field theory (DMFT)! |!1998: dynamical cluster approximation (DCA)! |!2000 - 2001: cellular dynamical mean-field theory (C-DMFT)! |!2003: variational cluster approach (VCA)! |!2004: periodized cellular dynamical mean-field theory (PC-DMFT)! |!2006: periodic cluster-perturbation theory! |

lattice models

= +

1.4 Michael Potthoff

2 The cluster approach

Tiling the lattice into small clusters

We start with the second question and consider a simple non-interacting system given by

H0

=

X

ij�

tijc†i�cj� = H

0

(t) . (2)

Here, c†i� creates an electron with spin � =", # at the site i of a D-dimensional lattice, and tijare the (spin-independent) hopping parameters which are also considered as the elements of thehopping matrix t. Furthermore,

H 00

=

X

ij�

t0ijc†i�cj� = H

0

(t0) , (3)

denotes the Hamiltonian of the system with decoupled clusters (see Fig. 1 and take H1

= 0). IfL is the number of lattice sites in the original lattice model H

0

and Lc

is the number of sites in anindividual cluster, there are L/L

c

decoupled clusters. We assume that all clusters are identical.In terms of hopping matrices, we have

t = t0 + V (4)

where V is the inter-cluster hopping.Consider the resolvent of the hopping matrix, i.e. the Green’s function

G0

(!) =1

! + µ� t. (5)

Here, ! is a complex frequency (units with ~ = 1 are used). We have also introduced thechemical potential µ (which is not important here but used later). Furthermore, we employ amatrix notation and write ! rather than !1 for short etc. Note the (· · · )�1 and 1/(· · · ) meansmatrix inversion.Having the Green’s function of the reference system at hand,

G00

(!) =1

! + µ� t0, (6)

how can be get the Green’s function of the original model? With some algebra, one easilyderives the equation

G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)

We see that using Green’s functions it is formally rather easy to couple a system of isolatedclusters.

Variational Cluster Approximation 1.5

Cluster perturbation theory

Actually, we are interested in interacting systems. For the single-orbital model H0

, the onlypossible local interaction is a Hubbard interaction of the form

H1

=

U

2

X

i�

ni�ni�� (9)

with ni� = c†i�ci� and where U is the interaction strength. Since H1

is completely local, theHamiltonian of the so-called “reference system” H 0

= H0

(t0)+H1

is obtained form the Hamil-tonian of the “original system” H = H

0

(t) + H1

by switching off the inter-cluster hoppingV .For a small cluster and likewise for a system of disconnected clusters, even for the interactingcase, it is comparatively simple to solve the problem exactly (by numerical means if necessary)while for the original lattice model this is a hard problem. One therefore cannot expect a simplerelation between the original and the reference system like Eq. (7). Nevertheless, as it is tootempting, we will write down

G(!) = G0(!) +G0

(!)V G(!) (10)

where now G and G0 are interacting Green’s functions. This is an equation that constitutesthe cluster-perturbation theory [17, 18]. It must be seen as an approximate way to compute theGreen’s function of the interacting model from the exact cluster Green’s function. In a waythe approximation is controlled by the size L

c

of the clusters in the reference system since forLc

! 1 one can expect the approximation to become exact. In fact, the CPT is not too badand has been successfully applied in a couple of problems, see Ref. [19] and references therein.

Green’s function and exact diagonalization

Before proceeding with the interpretation of the CPT equation (10) which provides an approxi-mate expression for G(!), let us give the exact definition of the Green’s function for the inter-acting case. Its elements are defined as

Gij�(!) =

Z 1

�1dz

Aij�(z)

! � z, (11)

where ! is an arbitrary complex frequency and where

Aij�(z) =

Z 1

�1dt eiztAij�(t) (12)

is the single-particle spectral density whose Fourier transform

Aij�(t) =1

2⇡h[ci�(t), c†j�(0)]+i (13)

is given as the thermal expectation value of the anti-commutator of the annihilator with thecreator in the (grand-canonical) Heisenberg picture, e.g.

ci�(t) = ei(H�µN)tci�e�i(H�µN)t (14)

kinetic and potential energy

interaction energy

Why consider the Hubbard model ?!• generic many-body problem!• most simple setup for the

“correlation problem”

H = H0 +H1<latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit><latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit><latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit><latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit>

e.g.: Hubbard model

Hilbert space

= +

single site: dimension 4

L sites: dimension 4 = eL ln4 L.

using symmetries: and are conserved!

dimension !

for L=10: 63504 (half-filling)!for L=12: 853776!accessible by Krylov-space methods

✓LN"

◆✓LN#

◆N" N#

H = H0 +H1<latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit><latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit><latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit><latexit sha1_base64="fZjTR6bVuTDjnVgjPAxTs/AeCwE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIsgCGWmCLoRCm66rGAf0I4lk2ba0CQzJBmlDP0PNy4Uceu/uPNvzLSz0NYD93I4515yc4KYM21c99sprK1vbG4Vt0s7u3v7B+XDo7aOEkVoi0Q8Ut0Aa8qZpC3DDKfdWFEsAk47weQ28zuPVGkWyXszjakv8EiykBFsrPTQQDeoMXDRhe1eaVCuuFV3DrRKvJxUIEdzUP7qDyOSCCoN4VjrnufGxk+xMoxwOiv1E01jTCZ4RHuWSiyo9tP51TN0ZpUhCiNlSxo0V39vpFhoPRWBnRTYjPWyl4n/eb3EhNd+ymScGCrJ4qEw4chEKIsADZmixPCpJZgoZm9FZIwVJsYGlYXgLX95lbRrVc+teneXlXotj6MIJ3AK5+DBFdShAU1oAQEFz/AKb86T8+K8Ox+L0YKT7xzDHzifP4jsj98=</latexit>

e.g.: Hubbard model

quantum statistics

lattice model

| 0i =X

m1,m2,...,mL

cm1,...,mL |m1i ⌦ · · ·⌦ |mLi<latexit sha1_base64="oS0QVOuWSgVjktgzaYLIUPIHKd8=">AAACUHicbVFNaxsxEJ11P5I4bbNtjrmImkAPYdk1gfQSCPTSQw8OxHbAaxatLNvC+lik2YJZ+yf2klt/Ry89JCSyvSmNkwFpnt6bYaSnvJDCYRz/DhqvXr95u7O719x/9/7DQfjxU8+Z0jLeZUYae51Tx6XQvIsCJb8uLKcql7yfz76t9P5Pbp0w+grnBR8qOtFiLBhFT2XhZJF2nMji1FI9kZycp65UWaWy5ISorH1CoihaoR9Lwmr6H7Pwe1I3pgaF4i5lI4OuPpCFr6r1LGzFUbwO8hwkNWhBHZ0svElHhpWKa2SSOjdI4gKHFbUomOTLZlo6XlA2oxM+8FBTP3BYrQ1ZkmPPjMjYWL80kjX7f0dFlXNzlftKRXHqtrUV+ZI2KHH8dVgJXZTINdsMGpeSoCErd8lIWM5Qzj2gzAp/V8Km1FKG/g+a3oRk+8nPQa8dJXGUXJ62Ltq1HbtwBJ/hCyRwBhfwHTrQBQa/4A/cwl1wE/wN7hvBpvQxwyE8iUbzAUmdskw=</latexit><latexit sha1_base64="oS0QVOuWSgVjktgzaYLIUPIHKd8=">AAACUHicbVFNaxsxEJ11P5I4bbNtjrmImkAPYdk1gfQSCPTSQw8OxHbAaxatLNvC+lik2YJZ+yf2klt/Ry89JCSyvSmNkwFpnt6bYaSnvJDCYRz/DhqvXr95u7O719x/9/7DQfjxU8+Z0jLeZUYae51Tx6XQvIsCJb8uLKcql7yfz76t9P5Pbp0w+grnBR8qOtFiLBhFT2XhZJF2nMji1FI9kZycp65UWaWy5ISorH1CoihaoR9Lwmr6H7Pwe1I3pgaF4i5lI4OuPpCFr6r1LGzFUbwO8hwkNWhBHZ0svElHhpWKa2SSOjdI4gKHFbUomOTLZlo6XlA2oxM+8FBTP3BYrQ1ZkmPPjMjYWL80kjX7f0dFlXNzlftKRXHqtrUV+ZI2KHH8dVgJXZTINdsMGpeSoCErd8lIWM5Qzj2gzAp/V8Km1FKG/g+a3oRk+8nPQa8dJXGUXJ62Ltq1HbtwBJ/hCyRwBhfwHTrQBQa/4A/cwl1wE/wN7hvBpvQxwyE8iUbzAUmdskw=</latexit><latexit sha1_base64="oS0QVOuWSgVjktgzaYLIUPIHKd8=">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</latexit><latexit sha1_base64="oS0QVOuWSgVjktgzaYLIUPIHKd8=">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</latexit>

S=1/2 Heisenberg model compute the ground state?

coefficients cannot be stored for large L!

quantum statistics:!compute the partition function / free energy

F (T, x) = �T lnZ(T, x)<latexit sha1_base64="wjR+GqTCG87KwchBQ3UqS1tr55o=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFqKBlpgi6EQqCuKzQF7ZDyaRpDc1khiQjltqFv+LGhSJu/Q13/o3pdBbaeuDCyTn3knuPH3GmtON8W5mFxaXllexqbm19Y3PL3t6pqzCWhNZIyEPZ9LGinAla00xz2owkxYHPacMfXE78xj2VioWiqocR9QLcF6zHCNZG6th7V4Xq8cMRukAnqNrmAt0m746dd4pOAjRP3JTkIUWlY3+1uyGJAyo04ViplutE2hthqRnhdJxrx4pGmAxwn7YMFTigyhsl+4/RoVG6qBdKU0KjRP09McKBUsPAN50B1ndq1puI/3mtWPfOvRETUaypINOPejFHOkSTMFCXSUo0HxqCiWRmV0TusMREm8hyJgR39uR5Ui8VXafo3pzmy6U0jizswwEUwIUzKMM1VKAGBB7hGV7hzXqyXqx362PamrHSmV34A+vzB3Wbkx0=</latexit><latexit sha1_base64="wjR+GqTCG87KwchBQ3UqS1tr55o=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFqKBlpgi6EQqCuKzQF7ZDyaRpDc1khiQjltqFv+LGhSJu/Q13/o3pdBbaeuDCyTn3knuPH3GmtON8W5mFxaXllexqbm19Y3PL3t6pqzCWhNZIyEPZ9LGinAla00xz2owkxYHPacMfXE78xj2VioWiqocR9QLcF6zHCNZG6th7V4Xq8cMRukAnqNrmAt0m746dd4pOAjRP3JTkIUWlY3+1uyGJAyo04ViplutE2hthqRnhdJxrx4pGmAxwn7YMFTigyhsl+4/RoVG6qBdKU0KjRP09McKBUsPAN50B1ndq1puI/3mtWPfOvRETUaypINOPejFHOkSTMFCXSUo0HxqCiWRmV0TusMREm8hyJgR39uR5Ui8VXafo3pzmy6U0jizswwEUwIUzKMM1VKAGBB7hGV7hzXqyXqx362PamrHSmV34A+vzB3Wbkx0=</latexit><latexit sha1_base64="wjR+GqTCG87KwchBQ3UqS1tr55o=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFqKBlpgi6EQqCuKzQF7ZDyaRpDc1khiQjltqFv+LGhSJu/Q13/o3pdBbaeuDCyTn3knuPH3GmtON8W5mFxaXllexqbm19Y3PL3t6pqzCWhNZIyEPZ9LGinAla00xz2owkxYHPacMfXE78xj2VioWiqocR9QLcF6zHCNZG6th7V4Xq8cMRukAnqNrmAt0m746dd4pOAjRP3JTkIUWlY3+1uyGJAyo04ViplutE2hthqRnhdJxrx4pGmAxwn7YMFTigyhsl+4/RoVG6qBdKU0KjRP09McKBUsPAN50B1ndq1puI/3mtWPfOvRETUaypINOPejFHOkSTMFCXSUo0HxqCiWRmV0TusMREm8hyJgR39uR5Ui8VXafo3pzmy6U0jizswwEUwIUzKMM1VKAGBB7hGV7hzXqyXqx362PamrHSmV34A+vzB3Wbkx0=</latexit><latexit sha1_base64="wjR+GqTCG87KwchBQ3UqS1tr55o=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFqKBlpgi6EQqCuKzQF7ZDyaRpDc1khiQjltqFv+LGhSJu/Q13/o3pdBbaeuDCyTn3knuPH3GmtON8W5mFxaXllexqbm19Y3PL3t6pqzCWhNZIyEPZ9LGinAla00xz2owkxYHPacMfXE78xj2VioWiqocR9QLcF6zHCNZG6th7V4Xq8cMRukAnqNrmAt0m746dd4pOAjRP3JTkIUWlY3+1uyGJAyo04ViplutE2hthqRnhdJxrx4pGmAxwn7YMFTigyhsl+4/RoVG6qBdKU0KjRP09McKBUsPAN50B1ndq1puI/3mtWPfOvRETUaypINOPejFHOkSTMFCXSUo0HxqCiWRmV0TusMREm8hyJgR39uR5Ui8VXafo3pzmy6U0jizswwEUwIUzKMM1VKAGBB7hGV7hzXqyXqx362PamrHSmV34A+vzB3Wbkx0=</latexit>

Z(T, x) = tr exp(��H(x))<latexit sha1_base64="2Dkv+6xePxQRsQJxfcvVl+VdWZA=">AAACDXicbVA9SwNBEN2LXzF+nVraLEYhgRjugqCNELCxVDAazIWwt5nokr0PdufkwpE/YONfsbFQxNbezn/jJqbQxAcDj/dmmJnnx1JodJwvKzc3v7C4lF8urKyurW/Ym1tXOkoUhwaPZKSaPtMgRQgNFCihGStggS/h2u+fjvzre1BaROElDmJoB+w2FD3BGRqpY+/dlC4raZmeeIEfpRmqoVfxII1LB54PyOhZKS2XO3bRqTpj0FniTkiRTHDesT+9bsSTAELkkmndcp0Y2xlTKLiEYcFLNMSM99kttAwNWQC6nY2/GdJ9o3RpL1KmQqRj9fdExgKtB4FvOgOGd3raG4n/ea0Ee8ftTIRxghDyn0W9RFKM6Cga2hUKOMqBIYwrYW6l/I4pxtEEWDAhuNMvz5KrWtV1qu7FYbFem8SRJztkl5SIS45InZyRc9IgnDyQJ/JCXq1H69l6s95/WnPWZGab/IH18Q1jl5nL</latexit><latexit sha1_base64="2Dkv+6xePxQRsQJxfcvVl+VdWZA=">AAACDXicbVA9SwNBEN2LXzF+nVraLEYhgRjugqCNELCxVDAazIWwt5nokr0PdufkwpE/YONfsbFQxNbezn/jJqbQxAcDj/dmmJnnx1JodJwvKzc3v7C4lF8urKyurW/Ym1tXOkoUhwaPZKSaPtMgRQgNFCihGStggS/h2u+fjvzre1BaROElDmJoB+w2FD3BGRqpY+/dlC4raZmeeIEfpRmqoVfxII1LB54PyOhZKS2XO3bRqTpj0FniTkiRTHDesT+9bsSTAELkkmndcp0Y2xlTKLiEYcFLNMSM99kttAwNWQC6nY2/GdJ9o3RpL1KmQqRj9fdExgKtB4FvOgOGd3raG4n/ea0Ee8ftTIRxghDyn0W9RFKM6Cga2hUKOMqBIYwrYW6l/I4pxtEEWDAhuNMvz5KrWtV1qu7FYbFem8SRJztkl5SIS45InZyRc9IgnDyQJ/JCXq1H69l6s95/WnPWZGab/IH18Q1jl5nL</latexit><latexit sha1_base64="2Dkv+6xePxQRsQJxfcvVl+VdWZA=">AAACDXicbVA9SwNBEN2LXzF+nVraLEYhgRjugqCNELCxVDAazIWwt5nokr0PdufkwpE/YONfsbFQxNbezn/jJqbQxAcDj/dmmJnnx1JodJwvKzc3v7C4lF8urKyurW/Ym1tXOkoUhwaPZKSaPtMgRQgNFCihGStggS/h2u+fjvzre1BaROElDmJoB+w2FD3BGRqpY+/dlC4raZmeeIEfpRmqoVfxII1LB54PyOhZKS2XO3bRqTpj0FniTkiRTHDesT+9bsSTAELkkmndcp0Y2xlTKLiEYcFLNMSM99kttAwNWQC6nY2/GdJ9o3RpL1KmQqRj9fdExgKtB4FvOgOGd3raG4n/ea0Ee8ftTIRxghDyn0W9RFKM6Cga2hUKOMqBIYwrYW6l/I4pxtEEWDAhuNMvz5KrWtV1qu7FYbFem8SRJztkl5SIS45InZyRc9IgnDyQJ/JCXq1H69l6s95/WnPWZGab/IH18Q1jl5nL</latexit><latexit sha1_base64="2Dkv+6xePxQRsQJxfcvVl+VdWZA=">AAACDXicbVA9SwNBEN2LXzF+nVraLEYhgRjugqCNELCxVDAazIWwt5nokr0PdufkwpE/YONfsbFQxNbezn/jJqbQxAcDj/dmmJnnx1JodJwvKzc3v7C4lF8urKyurW/Ym1tXOkoUhwaPZKSaPtMgRQgNFCihGStggS/h2u+fjvzre1BaROElDmJoB+w2FD3BGRqpY+/dlC4raZmeeIEfpRmqoVfxII1LB54PyOhZKS2XO3bRqTpj0FniTkiRTHDesT+9bsSTAELkkmndcp0Y2xlTKLiEYcFLNMSM99kttAwNWQC6nY2/GdJ9o3RpL1KmQqRj9fdExgKtB4FvOgOGd3raG4n/ea0Ee8ftTIRxghDyn0W9RFKM6Cga2hUKOMqBIYwrYW6l/I4pxtEEWDAhuNMvz5KrWtV1qu7FYbFem8SRJztkl5SIS45InZyRc9IgnDyQJ/JCXq1H69l6s95/WnPWZGab/IH18Q1jl5nL</latexit>

gives access to thermodynamics!|mii = | " i , | # i

<latexit sha1_base64="WnmojLpdP4AvXBrhyWcGTmIh0Ao=">AAACKHicbVDLSgMxFM34rPU16tJNsAguSpkpgm7EghuXFewDOsOQSTNtaCYZkoylTPs5bvwVNyKKdOuXmLaz0NYDFw7n3HuTe8KEUaUdZ2qtrW9sbm0Xdoq7e/sHh/bRcVOJVGLSwIIJ2Q6RIoxy0tBUM9JOJEFxyEgrHNzN/NYTkYoK/qhHCfFj1OM0ohhpIwX27RjGAfUk4j1G4A0ce2mCpBRDrwxz1bCyKWN1xZAvmYFdcirOHHCVuDkpgRz1wH43a3AaE64xQ0p1XCfRfoakppiRSdFLFUkQHqAe6RjKUUyUn80PncBzo3RhJKQpruFc/T2RoVipURyazhjpvlr2ZuJ/XifV0bWfUZ6kmnC8eChKGdQCzlKDXSoJ1mxkCMKSmr9C3EcSYW2yLZoQ3OWTV0mzWnGdivtwWapV8zgK4BScgQvggitQA/egDhoAg2fwCj7Ap/VivVlf1nTRumblMyfgD6zvH0J/pW0=</latexit><latexit sha1_base64="WnmojLpdP4AvXBrhyWcGTmIh0Ao=">AAACKHicbVDLSgMxFM34rPU16tJNsAguSpkpgm7EghuXFewDOsOQSTNtaCYZkoylTPs5bvwVNyKKdOuXmLaz0NYDFw7n3HuTe8KEUaUdZ2qtrW9sbm0Xdoq7e/sHh/bRcVOJVGLSwIIJ2Q6RIoxy0tBUM9JOJEFxyEgrHNzN/NYTkYoK/qhHCfFj1OM0ohhpIwX27RjGAfUk4j1G4A0ce2mCpBRDrwxz1bCyKWN1xZAvmYFdcirOHHCVuDkpgRz1wH43a3AaE64xQ0p1XCfRfoakppiRSdFLFUkQHqAe6RjKUUyUn80PncBzo3RhJKQpruFc/T2RoVipURyazhjpvlr2ZuJ/XifV0bWfUZ6kmnC8eChKGdQCzlKDXSoJ1mxkCMKSmr9C3EcSYW2yLZoQ3OWTV0mzWnGdivtwWapV8zgK4BScgQvggitQA/egDhoAg2fwCj7Ap/VivVlf1nTRumblMyfgD6zvH0J/pW0=</latexit><latexit sha1_base64="WnmojLpdP4AvXBrhyWcGTmIh0Ao=">AAACKHicbVDLSgMxFM34rPU16tJNsAguSpkpgm7EghuXFewDOsOQSTNtaCYZkoylTPs5bvwVNyKKdOuXmLaz0NYDFw7n3HuTe8KEUaUdZ2qtrW9sbm0Xdoq7e/sHh/bRcVOJVGLSwIIJ2Q6RIoxy0tBUM9JOJEFxyEgrHNzN/NYTkYoK/qhHCfFj1OM0ohhpIwX27RjGAfUk4j1G4A0ce2mCpBRDrwxz1bCyKWN1xZAvmYFdcirOHHCVuDkpgRz1wH43a3AaE64xQ0p1XCfRfoakppiRSdFLFUkQHqAe6RjKUUyUn80PncBzo3RhJKQpruFc/T2RoVipURyazhjpvlr2ZuJ/XifV0bWfUZ6kmnC8eChKGdQCzlKDXSoJ1mxkCMKSmr9C3EcSYW2yLZoQ3OWTV0mzWnGdivtwWapV8zgK4BScgQvggitQA/egDhoAg2fwCj7Ap/VivVlf1nTRumblMyfgD6zvH0J/pW0=</latexit><latexit sha1_base64="WnmojLpdP4AvXBrhyWcGTmIh0Ao=">AAACKHicbVDLSgMxFM34rPU16tJNsAguSpkpgm7EghuXFewDOsOQSTNtaCYZkoylTPs5bvwVNyKKdOuXmLaz0NYDFw7n3HuTe8KEUaUdZ2qtrW9sbm0Xdoq7e/sHh/bRcVOJVGLSwIIJ2Q6RIoxy0tBUM9JOJEFxyEgrHNzN/NYTkYoK/qhHCfFj1OM0ohhpIwX27RjGAfUk4j1G4A0ce2mCpBRDrwxz1bCyKWN1xZAvmYFdcirOHHCVuDkpgRz1wH43a3AaE64xQ0p1XCfRfoakppiRSdFLFUkQHqAe6RjKUUyUn80PncBzo3RhJKQpruFc/T2RoVipURyazhjpvlr2ZuJ/XifV0bWfUZ6kmnC8eChKGdQCzlKDXSoJ1mxkCMKSmr9C3EcSYW2yLZoQ3OWTV0mzWnGdivtwWapV8zgK4BScgQvggitQA/egDhoAg2fwCj7Ap/VivVlf1nTRumblMyfgD6zvH0J/pW0=</latexit>

DMFT: provides the thermodynamics (and excitations)

H =X

i,j

JijSiSj �BX

i

Siz

<latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit><latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit><latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit><latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit>

correlated lattice models

DMFT: aims at strongly correlated lattice fermion models !

phenomena:Kondo effect, Mott transition, collective order of spin, charge, orbital degrees of freedom, superconductivity, heavy-fermion behavior, etc.

Heisenberg model, J=0 Z = tr exp

�B

X

i

Siz

!=

X

m1,...,mL

exp

�B

X

i

mi

!

=

X

m1,...,mL

Y

i

exp (�Bmi) =

Y

i

X

mi

exp (�Bmi)

= (exp(�B/2) + exp(��B/2))L = Z1 · · ·ZL = ZL1

<latexit sha1_base64="DePKlUCDmcN9SDe8HIu1lqkUMeA=">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</latexit><latexit sha1_base64="DePKlUCDmcN9SDe8HIu1lqkUMeA=">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</latexit><latexit sha1_base64="DePKlUCDmcN9SDe8HIu1lqkUMeA=">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</latexit><latexit sha1_base64="DePKlUCDmcN9SDe8HIu1lqkUMeA=">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</latexit>

noninteracting system can be treated easily

H = �BX

i

Siz

<latexit sha1_base64="C71rWdtFv2G1dD67VLx4+M1eu9Q=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEN5akCLoRim66rGgf0IYwmU7boTNJmJkIbSj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BzJnSjvNt5VZW19Y38puFre2d3T17/6ChokQSWicRj2QrwIpyFtK6ZprTViwpFgGnzWB4O/Wbj1QqFoUPehRTT+B+yHqMYG0k3z6qomt0jm46KhE+Q/d+ysYT3y46JWcGtEzcjBQhQ823vzrdiCSChppwrFTbdWLtpVhqRjidFDqJojEmQ9ynbUNDLKjy0tn1E3RqlC7qRdJUqNFM/T2RYqHUSASmU2A9UIveVPzPaye6d+WlLIwTTUMyX9RLONIRmkaBukxSovnIEEwkM7ciMsASE20CK5gQ3MWXl0mjXHKdknt3UayUszjycAwncAYuXEIFqlCDOhAYwzO8wpv1ZL1Y79bHvDVnZTOH8AfW5w9nP5Pd</latexit><latexit sha1_base64="C71rWdtFv2G1dD67VLx4+M1eu9Q=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEN5akCLoRim66rGgf0IYwmU7boTNJmJkIbSj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BzJnSjvNt5VZW19Y38puFre2d3T17/6ChokQSWicRj2QrwIpyFtK6ZprTViwpFgGnzWB4O/Wbj1QqFoUPehRTT+B+yHqMYG0k3z6qomt0jm46KhE+Q/d+ysYT3y46JWcGtEzcjBQhQ823vzrdiCSChppwrFTbdWLtpVhqRjidFDqJojEmQ9ynbUNDLKjy0tn1E3RqlC7qRdJUqNFM/T2RYqHUSASmU2A9UIveVPzPaye6d+WlLIwTTUMyX9RLONIRmkaBukxSovnIEEwkM7ciMsASE20CK5gQ3MWXl0mjXHKdknt3UayUszjycAwncAYuXEIFqlCDOhAYwzO8wpv1ZL1Y79bHvDVnZTOH8AfW5w9nP5Pd</latexit><latexit sha1_base64="C71rWdtFv2G1dD67VLx4+M1eu9Q=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEN5akCLoRim66rGgf0IYwmU7boTNJmJkIbSj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BzJnSjvNt5VZW19Y38puFre2d3T17/6ChokQSWicRj2QrwIpyFtK6ZprTViwpFgGnzWB4O/Wbj1QqFoUPehRTT+B+yHqMYG0k3z6qomt0jm46KhE+Q/d+ysYT3y46JWcGtEzcjBQhQ823vzrdiCSChppwrFTbdWLtpVhqRjidFDqJojEmQ9ynbUNDLKjy0tn1E3RqlC7qRdJUqNFM/T2RYqHUSASmU2A9UIveVPzPaye6d+WlLIwTTUMyX9RLONIRmkaBukxSovnIEEwkM7ciMsASE20CK5gQ3MWXl0mjXHKdknt3UayUszjycAwncAYuXEIFqlCDOhAYwzO8wpv1ZL1Y79bHvDVnZTOH8AfW5w9nP5Pd</latexit><latexit sha1_base64="C71rWdtFv2G1dD67VLx4+M1eu9Q=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEN5akCLoRim66rGgf0IYwmU7boTNJmJkIbSj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BzJnSjvNt5VZW19Y38puFre2d3T17/6ChokQSWicRj2QrwIpyFtK6ZprTViwpFgGnzWB4O/Wbj1QqFoUPehRTT+B+yHqMYG0k3z6qomt0jm46KhE+Q/d+ysYT3y46JWcGtEzcjBQhQ823vzrdiCSChppwrFTbdWLtpVhqRjidFDqJojEmQ9ynbUNDLKjy0tn1E3RqlC7qRdJUqNFM/T2RYqHUSASmU2A9UIveVPzPaye6d+WlLIwTTUMyX9RLONIRmkaBukxSovnIEEwkM7ciMsASE20CK5gQ3MWXl0mjXHKdknt3UayUszjycAwncAYuXEIFqlCDOhAYwzO8wpv1ZL1Y79bHvDVnZTOH8AfW5w9nP5Pd</latexit>

no correlationsmagnetic phase diagram??

mean-field approach

single-site mean-field theory

• mean field represents the environment!• should be determined (at best) from the solution of the lattice model!• pragmatically from the solution of HMF, i.e.: BMF = -2qJ<Siz>!• requires selfconsistent solution!

MFT: impurity model selfconsistently embedded in a bath

H =X

i,j

JijSiSj �BX

i

Siz

<latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit><latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit><latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit><latexit sha1_base64="NtU4Qa064O+e9YoX+q0ukZwsOlM=">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</latexit>

HMF = �BX

i

Siz �BMF

X

i

Siz

<latexit sha1_base64="yCykpJaXO5MChDJCzU/PFU2x7iM=">AAACIXicbVDLSsNAFJ3UV62vqEs3g0VwY0mKYDdCUZBuhIr2AU0Ik+mkHTp5MDMRasivuPFX3LhQpDvxZ5y0AW3rgQuHc+7l3nvciFEhDeNLK6ysrq1vFDdLW9s7u3v6/kFbhDHHpIVDFvKuiwRhNCAtSSUj3YgT5LuMdNzRdeZ3HgkXNAwe5Dgito8GAfUoRlJJjl5rOIklPHh7k8JLeAavLBH7DoX3TkKf0kz49ecsRy8bFWMKuEzMnJRBjqajT6x+iGOfBBIzJETPNCJpJ4hLihlJS1YsSITwCA1IT9EA+UTYyfTDFJ4opQ+9kKsKJJyqfycS5Asx9l3V6SM5FIteJv7n9WLp1eyEBlEsSYBni7yYQRnCLC7Yp5xgycaKIMypuhXiIeIISxVqSYVgLr68TNrVimlUzLvzcr2ax1EER+AYnAITXIA6aIAmaAEMnsEreAcf2ov2pn1qk1lrQctnDsEctO8fRV2huA==</latexit><latexit sha1_base64="yCykpJaXO5MChDJCzU/PFU2x7iM=">AAACIXicbVDLSsNAFJ3UV62vqEs3g0VwY0mKYDdCUZBuhIr2AU0Ik+mkHTp5MDMRasivuPFX3LhQpDvxZ5y0AW3rgQuHc+7l3nvciFEhDeNLK6ysrq1vFDdLW9s7u3v6/kFbhDHHpIVDFvKuiwRhNCAtSSUj3YgT5LuMdNzRdeZ3HgkXNAwe5Dgito8GAfUoRlJJjl5rOIklPHh7k8JLeAavLBH7DoX3TkKf0kz49ecsRy8bFWMKuEzMnJRBjqajT6x+iGOfBBIzJETPNCJpJ4hLihlJS1YsSITwCA1IT9EA+UTYyfTDFJ4opQ+9kKsKJJyqfycS5Asx9l3V6SM5FIteJv7n9WLp1eyEBlEsSYBni7yYQRnCLC7Yp5xgycaKIMypuhXiIeIISxVqSYVgLr68TNrVimlUzLvzcr2ax1EER+AYnAITXIA6aIAmaAEMnsEreAcf2ov2pn1qk1lrQctnDsEctO8fRV2huA==</latexit><latexit sha1_base64="yCykpJaXO5MChDJCzU/PFU2x7iM=">AAACIXicbVDLSsNAFJ3UV62vqEs3g0VwY0mKYDdCUZBuhIr2AU0Ik+mkHTp5MDMRasivuPFX3LhQpDvxZ5y0AW3rgQuHc+7l3nvciFEhDeNLK6ysrq1vFDdLW9s7u3v6/kFbhDHHpIVDFvKuiwRhNCAtSSUj3YgT5LuMdNzRdeZ3HgkXNAwe5Dgito8GAfUoRlJJjl5rOIklPHh7k8JLeAavLBH7DoX3TkKf0kz49ecsRy8bFWMKuEzMnJRBjqajT6x+iGOfBBIzJETPNCJpJ4hLihlJS1YsSITwCA1IT9EA+UTYyfTDFJ4opQ+9kKsKJJyqfycS5Asx9l3V6SM5FIteJv7n9WLp1eyEBlEsSYBni7yYQRnCLC7Yp5xgycaKIMypuhXiIeIISxVqSYVgLr68TNrVimlUzLvzcr2ax1EER+AYnAITXIA6aIAmaAEMnsEreAcf2ov2pn1qk1lrQctnDsEctO8fRV2huA==</latexit><latexit sha1_base64="yCykpJaXO5MChDJCzU/PFU2x7iM=">AAACIXicbVDLSsNAFJ3UV62vqEs3g0VwY0mKYDdCUZBuhIr2AU0Ik+mkHTp5MDMRasivuPFX3LhQpDvxZ5y0AW3rgQuHc+7l3nvciFEhDeNLK6ysrq1vFDdLW9s7u3v6/kFbhDHHpIVDFvKuiwRhNCAtSSUj3YgT5LuMdNzRdeZ3HgkXNAwe5Dgito8GAfUoRlJJjl5rOIklPHh7k8JLeAavLBH7DoX3TkKf0kz49ecsRy8bFWMKuEzMnJRBjqajT6x+iGOfBBIzJETPNCJpJ4hLihlJS1YsSITwCA1IT9EA+UTYyfTDFJ4opQ+9kKsKJJyqfycS5Asx9l3V6SM5FIteJv7n9WLp1eyEBlEsSYBni7yYQRnCLC7Yp5xgycaKIMypuhXiIeIISxVqSYVgLr68TNrVimlUzLvzcr2ax1EER+AYnAITXIA6aIAmaAEMnsEreAcf2ov2pn1qk1lrQctnDsEctO8fRV2huA==</latexit>

lattice model

fluctuating local field constant mean field

MFT

dynamical mean-field theory

• much more complicated as compared to Weiss MFTself-consistency equation formulated in terms of Green’s functions!

• optimal mean-field theory for lattice-fermion models (e.g. Hubbard model)point of orientation in the landscape of various MFT’s!

• no internal inconsistencies, not restricted to a certain parameter range (weak / strong interaction etc.), nonperturbative theory!

• exact theory in the (carefully defined) limit of infinite dimensions but usually applied for D < o as an approximation!

• is able to describe spontaneous symmetry breaking (magnetism, SC, …)!• can be derived in various way!• and highly successful, including applications to real materials (LDA+DMFT)

o

UtDMFT

but …

• short-range correlations are treated insufficiently!• no feedback of nonlocal two-particle (e.g. magnetic) correlation on the

Green’s function!• consequence: e.g. qualitatively wrong phase diagram in two dimensions!• no symmetry-broken phases with nonlocal order parameters

including unconventional (d-wave) superconductivity!• incorrect critical behavior close to second-order phase transitions!• violation of exact identities and sum rules, violation of the Mermin-Wagner

theorem, etc.!• DMFT (if applied to finite-D models) is approximate!

WANTED: systematic route from DMFT to the exact solution

UtDMFT

cluster mean-field approach

lattice model single-site mean-field theory

cluster mean-field theory

self-consistent embedding of a cluster with Lc sites in a bath!

systematic, exact for infinite Lc!computational effort expected to increase strongly with Lc!

GEOMETRY

lattice and reciprocal lattice

R =DX

↵=1

i↵a↵

<latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit><latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit><latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit><latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit>

G =DX

�=1

j�b�<latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="feYzCSbqj61LrrcowietRi1DSP8=">AAAB5HicbVDLSgNBEOz1GddX9OplMAiewq4XPQpePEYwD0hCmJ3tTYbMzi4zvUJY8gMevIh49Zu8+TdOHqAmFjQUVd10d0W5kpaC4Mvb2Nza3tmt7Pn7B/7h0XH1pGWzwghsikxlphNxi0pqbJIkhZ3cIE8jhe1ofDfz209orMz0I01y7Kd8qGUiBScnNQbVWlAP5mDrJFySGiwxqH724kwUKWoSilvbDYOc+iU3JIXCqd8rLOZcjPkQu45qnqLtl/Mzp+zCKTFLMuNKE5urvydKnlo7SSPXmXIa2VVvJv7ndQtKbvql1HlBqMViUVIoRhmb/cxiaVCQmjjChZHuViZG3HBBLhnfZRCufrxOWlf1MKiHDz9hQAXO4BwuIYRruIV7aEATBMTwDK/eyHvx3rz3ReOGt5w4hT/wPr4BGDqLAg==</latexit><latexit sha1_base64="iC3FV0sIqBK97Y3uXBKAMCLRIj0=">AAACKXicbVDLSgMxFL3js9ZX1aWbYBFclRk3uikUFHRZwT6gHYdMmmljM5MhyQhlmN9x46+4UVDUrT9ipi1YWw+EnJx7Lrn3+DFnStv2p7W0vLK6tl7YKG5ube/slvb2m0okktAGEVzIto8V5SyiDc00p+1YUhz6nLb84UVebz1QqZiIbvUopm6I+xELGMHaSF6plnZ9wXtqFJoLXWWo2lVJ6BmValx1srv0MkP33viJZqypn01Er1S2K/YYaJE4U1KGKepe6bXbEyQJaaQJx0p1HDvWboqlZoTTrNhNFI0xGeI+7Rga4ZAqNx1vmqFjo/RQIKQ5kUZjdbYjxaHKBzTOEOuBmq/l4n+1TqKDczdlUZxoGpHJR0HCkRYojw31mKRE85EhmEhmZkVkgCUm2oRbNCE48ysvkuZpxbErzs1vGlCAQziCE3DgDGpwDXVoAIFHeIY3eLeerBfrw/qaWJesac8B/IH1/QP3lqdB</latexit><latexit sha1_base64="iC3FV0sIqBK97Y3uXBKAMCLRIj0=">AAACKXicbVDLSgMxFL3js9ZX1aWbYBFclRk3uikUFHRZwT6gHYdMmmljM5MhyQhlmN9x46+4UVDUrT9ipi1YWw+EnJx7Lrn3+DFnStv2p7W0vLK6tl7YKG5ube/slvb2m0okktAGEVzIto8V5SyiDc00p+1YUhz6nLb84UVebz1QqZiIbvUopm6I+xELGMHaSF6plnZ9wXtqFJoLXWWo2lVJ6BmValx1srv0MkP33viJZqypn01Er1S2K/YYaJE4U1KGKepe6bXbEyQJaaQJx0p1HDvWboqlZoTTrNhNFI0xGeI+7Rga4ZAqNx1vmqFjo/RQIKQ5kUZjdbYjxaHKBzTOEOuBmq/l4n+1TqKDczdlUZxoGpHJR0HCkRYojw31mKRE85EhmEhmZkVkgCUm2oRbNCE48ysvkuZpxbErzs1vGlCAQziCE3DgDGpwDXVoAIFHeIY3eLeerBfrw/qaWJesac8B/IH1/QP3lqdB</latexit><latexit sha1_base64="Fh7k5r2yFq73WxqVPul8HWDl+90=">AAACKXicbVBNS8MwGE79nPOr6tFLcAieRruLXgYDBT1OcB+wzpKm6RaXNiVJhVH6d7z4V7woKOrVP2La9TA3Hwh58rzPS9738WJGpbKsL2NldW19Y7OyVd3e2d3bNw8Ou5InApMO5oyLvockYTQiHUUVI/1YEBR6jPS8yWVe7z0SISmP7tQ0JsMQjSIaUIyUllyzlToeZ76chvqC1xlsOjIJXa0ShZp2dp9eZfDBLZ5wzpp62Ux0zZpVtwrAZWKXpAZKtF3zzfE5TkISKcyQlAPbitUwRUJRzEhWdRJJYoQnaEQGmkYoJHKYFptm8FQrPgy40CdSsFDnO1IUynxA7QyRGsvFWi7+VxskKrgYpjSKE0UiPPsoSBhUHOaxQZ8KghWbaoKwoHpWiMdIIKx0uFUdgr248jLpNuq2VbdvrVqrUcZRAcfgBJwBG5yDFrgBbdABGDyBF/AOPoxn49X4NL5n1hWj7DkCf2D8/AIaL6ez</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit>

exp(iGR) = 1

<latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit><latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit><latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit><latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit>

a↵b� = 2⇡�↵�<latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit><latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit><latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit><latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit>

VCVRC = (2⇡)D<latexit sha1_base64="gQ8XRWk4QqqZ2IdV4MvmHx2Lzpo=">AAACCnicbZC7SgNBFIbPxluMt1VLm9EgxCbsBkEbIRALyyjmAtk1zE5mkyGzF2ZmhbCktvFVbCwUsfUJ7HwbJ8kKmvjDwMd/zuHM+b2YM6ks68vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vGFtUm/dUyFZFN6qUUzdAPdD5jOClba65mGzmzoEc1Qbox+80XyBShUnZid3l12zaJWtqdAi2BkUIVO9a346vYgkAQ0V4VjKjm3Fyk2xUIxwOi44iaQxJkPcpx2NIQ6odNPpKWN0rJ0e8iOhX6jQ1P09keJAylHg6c4Aq4Gcr03M/2qdRPnnbsrCOFE0JLNFfsKRitAkF9RjghLFRxowEUz/FZEBFpgonV5Bh2DPn7wIzUrZtsr29WmxWsniyMMBHEEJbDiDKlxBHRpA4AGe4AVejUfj2Xgz3metOSOb2Yc/Mj6+AeSjmGY=</latexit><latexit sha1_base64="gQ8XRWk4QqqZ2IdV4MvmHx2Lzpo=">AAACCnicbZC7SgNBFIbPxluMt1VLm9EgxCbsBkEbIRALyyjmAtk1zE5mkyGzF2ZmhbCktvFVbCwUsfUJ7HwbJ8kKmvjDwMd/zuHM+b2YM6ks68vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vGFtUm/dUyFZFN6qUUzdAPdD5jOClba65mGzmzoEc1Qbox+80XyBShUnZid3l12zaJWtqdAi2BkUIVO9a346vYgkAQ0V4VjKjm3Fyk2xUIxwOi44iaQxJkPcpx2NIQ6odNPpKWN0rJ0e8iOhX6jQ1P09keJAylHg6c4Aq4Gcr03M/2qdRPnnbsrCOFE0JLNFfsKRitAkF9RjghLFRxowEUz/FZEBFpgonV5Bh2DPn7wIzUrZtsr29WmxWsniyMMBHEEJbDiDKlxBHRpA4AGe4AVejUfj2Xgz3metOSOb2Yc/Mj6+AeSjmGY=</latexit><latexit sha1_base64="gQ8XRWk4QqqZ2IdV4MvmHx2Lzpo=">AAACCnicbZC7SgNBFIbPxluMt1VLm9EgxCbsBkEbIRALyyjmAtk1zE5mkyGzF2ZmhbCktvFVbCwUsfUJ7HwbJ8kKmvjDwMd/zuHM+b2YM6ks68vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vGFtUm/dUyFZFN6qUUzdAPdD5jOClba65mGzmzoEc1Qbox+80XyBShUnZid3l12zaJWtqdAi2BkUIVO9a346vYgkAQ0V4VjKjm3Fyk2xUIxwOi44iaQxJkPcpx2NIQ6odNPpKWN0rJ0e8iOhX6jQ1P09keJAylHg6c4Aq4Gcr03M/2qdRPnnbsrCOFE0JLNFfsKRitAkF9RjghLFRxowEUz/FZEBFpgonV5Bh2DPn7wIzUrZtsr29WmxWsniyMMBHEEJbDiDKlxBHRpA4AGe4AVejUfj2Xgz3metOSOb2Yc/Mj6+AeSjmGY=</latexit><latexit sha1_base64="gQ8XRWk4QqqZ2IdV4MvmHx2Lzpo=">AAACCnicbZC7SgNBFIbPxluMt1VLm9EgxCbsBkEbIRALyyjmAtk1zE5mkyGzF2ZmhbCktvFVbCwUsfUJ7HwbJ8kKmvjDwMd/zuHM+b2YM6ks68vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vGFtUm/dUyFZFN6qUUzdAPdD5jOClba65mGzmzoEc1Qbox+80XyBShUnZid3l12zaJWtqdAi2BkUIVO9a346vYgkAQ0V4VjKjm3Fyk2xUIxwOi44iaQxJkPcpx2NIQ6odNPpKWN0rJ0e8iOhX6jQ1P09keJAylHg6c4Aq4Gcr03M/2qdRPnnbsrCOFE0JLNFfsKRitAkF9RjghLFRxowEUz/FZEBFpgonV5Bh2DPn7wIzUrZtsr29WmxWsniyMMBHEEJbDiDKlxBHRpA4AGe4AVejUfj2Xgz3metOSOb2Yc/Mj6+AeSjmGY=</latexit>

a1<latexit sha1_base64="7d82cpLEHfoMPQJLmtLH3CJ3FNg=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRwS4LblxWsA9ohyGTSdvQTDIkmUIZ+iduXCji1j9x59+YaWehrQdCDufcS05OlHKmjed9O5Wt7Z3dvep+7eDw6PjEPT3rapkpQjtEcqn6EdaUM0E7hhlO+6miOIk47UXT+8LvzajSTIonM09pkOCxYCNGsLFS6LrDSPJYzxN75XgR+qFb9xreEmiT+CWpQ4l26H4NY0myhApDONZ64HupCXKsDCOcLmrDTNMUkyke04GlAidUB/ky+QJdWSVGI6nsEQYt1d8bOU50Ec5OJthM9LpXiP95g8yMmkHORJoZKsjqoVHGkZGoqAHFTFFi+NwSTBSzWRGZYIWJsWXVbAn++pc3Sfem4XsN//G23mqWdVThAi7hGny4gxY8QBs6QGAGz/AKb07uvDjvzsdqtOKUO+fwB87nD9Wvk70=</latexit><latexit sha1_base64="7d82cpLEHfoMPQJLmtLH3CJ3FNg=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRwS4LblxWsA9ohyGTSdvQTDIkmUIZ+iduXCji1j9x59+YaWehrQdCDufcS05OlHKmjed9O5Wt7Z3dvep+7eDw6PjEPT3rapkpQjtEcqn6EdaUM0E7hhlO+6miOIk47UXT+8LvzajSTIonM09pkOCxYCNGsLFS6LrDSPJYzxN75XgR+qFb9xreEmiT+CWpQ4l26H4NY0myhApDONZ64HupCXKsDCOcLmrDTNMUkyke04GlAidUB/ky+QJdWSVGI6nsEQYt1d8bOU50Ec5OJthM9LpXiP95g8yMmkHORJoZKsjqoVHGkZGoqAHFTFFi+NwSTBSzWRGZYIWJsWXVbAn++pc3Sfem4XsN//G23mqWdVThAi7hGny4gxY8QBs6QGAGz/AKb07uvDjvzsdqtOKUO+fwB87nD9Wvk70=</latexit><latexit sha1_base64="7d82cpLEHfoMPQJLmtLH3CJ3FNg=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRwS4LblxWsA9ohyGTSdvQTDIkmUIZ+iduXCji1j9x59+YaWehrQdCDufcS05OlHKmjed9O5Wt7Z3dvep+7eDw6PjEPT3rapkpQjtEcqn6EdaUM0E7hhlO+6miOIk47UXT+8LvzajSTIonM09pkOCxYCNGsLFS6LrDSPJYzxN75XgR+qFb9xreEmiT+CWpQ4l26H4NY0myhApDONZ64HupCXKsDCOcLmrDTNMUkyke04GlAidUB/ky+QJdWSVGI6nsEQYt1d8bOU50Ec5OJthM9LpXiP95g8yMmkHORJoZKsjqoVHGkZGoqAHFTFFi+NwSTBSzWRGZYIWJsWXVbAn++pc3Sfem4XsN//G23mqWdVThAi7hGny4gxY8QBs6QGAGz/AKb07uvDjvzsdqtOKUO+fwB87nD9Wvk70=</latexit><latexit sha1_base64="7d82cpLEHfoMPQJLmtLH3CJ3FNg=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRwS4LblxWsA9ohyGTSdvQTDIkmUIZ+iduXCji1j9x59+YaWehrQdCDufcS05OlHKmjed9O5Wt7Z3dvep+7eDw6PjEPT3rapkpQjtEcqn6EdaUM0E7hhlO+6miOIk47UXT+8LvzajSTIonM09pkOCxYCNGsLFS6LrDSPJYzxN75XgR+qFb9xreEmiT+CWpQ4l26H4NY0myhApDONZ64HupCXKsDCOcLmrDTNMUkyke04GlAidUB/ky+QJdWSVGI6nsEQYt1d8bOU50Ec5OJthM9LpXiP95g8yMmkHORJoZKsjqoVHGkZGoqAHFTFFi+NwSTBSzWRGZYIWJsWXVbAn++pc3Sfem4XsN//G23mqWdVThAi7hGny4gxY8QBs6QGAGz/AKb07uvDjvzsdqtOKUO+fwB87nD9Wvk70=</latexit>

a2<latexit sha1_base64="Jlle3yAmka55K0lyhBmvzafohjw=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFwZznO/5dsNp+ksgTaJW5IGlOj69tc4jEkaUaEJx0qNXCfRXoalZoTTvDZOFU0wmeMpHRkqcESVly2z5+jSKCGaxNIcodFS/b2R4UgV6cxkhPVMrXuF+J83SvWk7WVMJKmmgqwemqQc6RgVRaCQSUo0XxiCiWQmKyIzLDHRpq6aKcFd//Im6beartN0768bnXZZRxXO4QKuwIUb6MAddKEHBJ7gGV7hzcqtF+vd+liNVqxy5wz+wPr8AaiGlMo=</latexit><latexit sha1_base64="Jlle3yAmka55K0lyhBmvzafohjw=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFwZznO/5dsNp+ksgTaJW5IGlOj69tc4jEkaUaEJx0qNXCfRXoalZoTTvDZOFU0wmeMpHRkqcESVly2z5+jSKCGaxNIcodFS/b2R4UgV6cxkhPVMrXuF+J83SvWk7WVMJKmmgqwemqQc6RgVRaCQSUo0XxiCiWQmKyIzLDHRpq6aKcFd//Im6beartN0768bnXZZRxXO4QKuwIUb6MAddKEHBJ7gGV7hzcqtF+vd+liNVqxy5wz+wPr8AaiGlMo=</latexit><latexit sha1_base64="Jlle3yAmka55K0lyhBmvzafohjw=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFwZznO/5dsNp+ksgTaJW5IGlOj69tc4jEkaUaEJx0qNXCfRXoalZoTTvDZOFU0wmeMpHRkqcESVly2z5+jSKCGaxNIcodFS/b2R4UgV6cxkhPVMrXuF+J83SvWk7WVMJKmmgqwemqQc6RgVRaCQSUo0XxiCiWQmKyIzLDHRpq6aKcFd//Im6beartN0768bnXZZRxXO4QKuwIUb6MAddKEHBJ7gGV7hzcqtF+vd+liNVqxy5wz+wPr8AaiGlMo=</latexit><latexit sha1_base64="Jlle3yAmka55K0lyhBmvzafohjw=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFwZznO/5dsNp+ksgTaJW5IGlOj69tc4jEkaUaEJx0qNXCfRXoalZoTTvDZOFU0wmeMpHRkqcESVly2z5+jSKCGaxNIcodFS/b2R4UgV6cxkhPVMrXuF+J83SvWk7WVMJKmmgqwemqQc6RgVRaCQSUo0XxiCiWQmKyIzLDHRpq6aKcFd//Im6beartN0768bnXZZRxXO4QKuwIUb6MAddKEHBJ7gGV7hzcqtF+vd+liNVqxy5wz+wPr8AaiGlMo=</latexit>

b2<latexit sha1_base64="3GKEYm/GGfsDDwdoJgc+Mr3bjW8=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFxZkOd+y7cbTtNZAm0StyQNKNH17a9xGJM0okITjpUauU6ivQxLzQineW2cKppgMsdTOjJU4IgqL1tmz9GlUUI0iaU5QqOl+nsjw5Eq0pnJCOuZWvcK8T9vlOpJ28uYSFJNBVk9NEk50jEqikAhk5RovjAEE8lMVkRmWGKiTV01U4K7/uVN0m81Xafp3l83Ou2yjiqcwwVcgQs30IE76EIPCDzBM7zCm5VbL9a79bEarVjlzhn8gfX5A6oOlMs=</latexit><latexit sha1_base64="3GKEYm/GGfsDDwdoJgc+Mr3bjW8=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFxZkOd+y7cbTtNZAm0StyQNKNH17a9xGJM0okITjpUauU6ivQxLzQineW2cKppgMsdTOjJU4IgqL1tmz9GlUUI0iaU5QqOl+nsjw5Eq0pnJCOuZWvcK8T9vlOpJ28uYSFJNBVk9NEk50jEqikAhk5RovjAEE8lMVkRmWGKiTV01U4K7/uVN0m81Xafp3l83Ou2yjiqcwwVcgQs30IE76EIPCDzBM7zCm5VbL9a79bEarVjlzhn8gfX5A6oOlMs=</latexit><latexit sha1_base64="3GKEYm/GGfsDDwdoJgc+Mr3bjW8=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFxZkOd+y7cbTtNZAm0StyQNKNH17a9xGJM0okITjpUauU6ivQxLzQineW2cKppgMsdTOjJU4IgqL1tmz9GlUUI0iaU5QqOl+nsjw5Eq0pnJCOuZWvcK8T9vlOpJ28uYSFJNBVk9NEk50jEqikAhk5RovjAEE8lMVkRmWGKiTV01U4K7/uVN0m81Xafp3l83Ou2yjiqcwwVcgQs30IE76EIPCDzBM7zCm5VbL9a79bEarVjlzhn8gfX5A6oOlMs=</latexit><latexit sha1_base64="3GKEYm/GGfsDDwdoJgc+Mr3bjW8=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmKYJcFNy4r2Ae0IUwm03boZBJmJmIJ+RU3LhRx64+482+ctFlo64FhDufcy5w5QcKZ0o7zbVW2tnd296r7tYPDo+MT+7TeV3EqCe2RmMdyGGBFORO0p5nmdJhIiqOA00Ewvy38wSOVisXiQS8S6kV4KtiEEayN5Nv1bBzEPFSLyFxZkOd+y7cbTtNZAm0StyQNKNH17a9xGJM0okITjpUauU6ivQxLzQineW2cKppgMsdTOjJU4IgqL1tmz9GlUUI0iaU5QqOl+nsjw5Eq0pnJCOuZWvcK8T9vlOpJ28uYSFJNBVk9NEk50jEqikAhk5RovjAEE8lMVkRmWGKiTV01U4K7/uVN0m81Xafp3l83Ou2yjiqcwwVcgQs30IE76EIPCDzBM7zCm5VbL9a79bEarVjlzhn8gfX5A6oOlMs=</latexit>

b1<latexit sha1_base64="V/qn+ErQ+6Lc8GM1Wo7FZwFhnUU=">AAAB+3icbVBPS8MwHP11/pvzX51HL8EheBqtCO448OJxgtuErZQ0TbewNC1JKo7Sr+LFgyJe/SLe/DamWw+6+SDk8d7vR15ekHKmtON8W7WNza3tnfpuY2//4PDIPm4OVJJJQvsk4Yl8CLCinAna10xz+pBKiuOA02Ewuyn94SOViiXiXs9T6sV4IljECNZG8u1mPg4SHqp5bK48KArf9e2W03YWQOvErUgLKvR8+2scJiSLqdCEY6VGrpNqL8dSM8Jp0RhniqaYzPCEjgwVOKbKyxfZC3RulBBFiTRHaLRQf2/kOFZlOjMZYz1Vq14p/ueNMh11vJyJNNNUkOVDUcaRTlBZBAqZpETzuSGYSGayIjLFEhNt6mqYEtzVL6+TwWXbddru3VWr26nqqMMpnMEFuHANXbiFHvSBwBM8wyu8WYX1Yr1bH8vRmlXtnMAfWJ8/qIqUyg==</latexit><latexit sha1_base64="V/qn+ErQ+6Lc8GM1Wo7FZwFhnUU=">AAAB+3icbVBPS8MwHP11/pvzX51HL8EheBqtCO448OJxgtuErZQ0TbewNC1JKo7Sr+LFgyJe/SLe/DamWw+6+SDk8d7vR15ekHKmtON8W7WNza3tnfpuY2//4PDIPm4OVJJJQvsk4Yl8CLCinAna10xz+pBKiuOA02Ewuyn94SOViiXiXs9T6sV4IljECNZG8u1mPg4SHqp5bK48KArf9e2W03YWQOvErUgLKvR8+2scJiSLqdCEY6VGrpNqL8dSM8Jp0RhniqaYzPCEjgwVOKbKyxfZC3RulBBFiTRHaLRQf2/kOFZlOjMZYz1Vq14p/ueNMh11vJyJNNNUkOVDUcaRTlBZBAqZpETzuSGYSGayIjLFEhNt6mqYEtzVL6+TwWXbddru3VWr26nqqMMpnMEFuHANXbiFHvSBwBM8wyu8WYX1Yr1bH8vRmlXtnMAfWJ8/qIqUyg==</latexit><latexit sha1_base64="V/qn+ErQ+6Lc8GM1Wo7FZwFhnUU=">AAAB+3icbVBPS8MwHP11/pvzX51HL8EheBqtCO448OJxgtuErZQ0TbewNC1JKo7Sr+LFgyJe/SLe/DamWw+6+SDk8d7vR15ekHKmtON8W7WNza3tnfpuY2//4PDIPm4OVJJJQvsk4Yl8CLCinAna10xz+pBKiuOA02Ewuyn94SOViiXiXs9T6sV4IljECNZG8u1mPg4SHqp5bK48KArf9e2W03YWQOvErUgLKvR8+2scJiSLqdCEY6VGrpNqL8dSM8Jp0RhniqaYzPCEjgwVOKbKyxfZC3RulBBFiTRHaLRQf2/kOFZlOjMZYz1Vq14p/ueNMh11vJyJNNNUkOVDUcaRTlBZBAqZpETzuSGYSGayIjLFEhNt6mqYEtzVL6+TwWXbddru3VWr26nqqMMpnMEFuHANXbiFHvSBwBM8wyu8WYX1Yr1bH8vRmlXtnMAfWJ8/qIqUyg==</latexit><latexit sha1_base64="V/qn+ErQ+6Lc8GM1Wo7FZwFhnUU=">AAAB+3icbVBPS8MwHP11/pvzX51HL8EheBqtCO448OJxgtuErZQ0TbewNC1JKo7Sr+LFgyJe/SLe/DamWw+6+SDk8d7vR15ekHKmtON8W7WNza3tnfpuY2//4PDIPm4OVJJJQvsk4Yl8CLCinAna10xz+pBKiuOA02Ewuyn94SOViiXiXs9T6sV4IljECNZG8u1mPg4SHqp5bK48KArf9e2W03YWQOvErUgLKvR8+2scJiSLqdCEY6VGrpNqL8dSM8Jp0RhniqaYzPCEjgwVOKbKyxfZC3RulBBFiTRHaLRQf2/kOFZlOjMZYz1Vq14p/ueNMh11vJyJNNNUkOVDUcaRTlBZBAqZpETzuSGYSGayIjLFEhNt6mqYEtzVL6+TwWXbddru3VWr26nqqMMpnMEFuHANXbiFHvSBwBM8wyu8WYX1Yr1bH8vRmlXtnMAfWJ8/qIqUyg==</latexit>

A1<latexit sha1_base64="P47SsFqzSSxa0RYDRpj/aiGYLv0=">AAAB+3icbVDLSsNAFL2pr1pftS7dDBbBVUlEsMuKG5cV7APaECaTSTt0MgkzE7GE/IobF4q49Ufc+TdO2iy09cAwh3PuZc4cP+FMadv+tiobm1vbO9Xd2t7+weFR/bjRV3EqCe2RmMdy6GNFORO0p5nmdJhIiiOf04E/uy38wSOVisXiQc8T6kZ4IljICNZG8uqNbOzHPFDzyFzZTZ57jldv2i17AbROnJI0oUTXq3+Ng5ikERWacKzUyLET7WZYakY4zWvjVNEEkxme0JGhAkdUudkie47OjRKgMJbmCI0W6u+NDEeqSGcmI6ynatUrxP+8UarDtpsxkaSaCrJ8KEw50jEqikABk5RoPjcEE8lMVkSmWGKiTV01U4Kz+uV10r9sOXbLub9qdtplHVU4hTO4AAeuoQN30IUeEHiCZ3iFNyu3Xqx362M5WrHKnRP4A+vzB3YClKk=</latexit><latexit sha1_base64="P47SsFqzSSxa0RYDRpj/aiGYLv0=">AAAB+3icbVDLSsNAFL2pr1pftS7dDBbBVUlEsMuKG5cV7APaECaTSTt0MgkzE7GE/IobF4q49Ufc+TdO2iy09cAwh3PuZc4cP+FMadv+tiobm1vbO9Xd2t7+weFR/bjRV3EqCe2RmMdy6GNFORO0p5nmdJhIiiOf04E/uy38wSOVisXiQc8T6kZ4IljICNZG8uqNbOzHPFDzyFzZTZ57jldv2i17AbROnJI0oUTXq3+Ng5ikERWacKzUyLET7WZYakY4zWvjVNEEkxme0JGhAkdUudkie47OjRKgMJbmCI0W6u+NDEeqSGcmI6ynatUrxP+8UarDtpsxkaSaCrJ8KEw50jEqikABk5RoPjcEE8lMVkSmWGKiTV01U4Kz+uV10r9sOXbLub9qdtplHVU4hTO4AAeuoQN30IUeEHiCZ3iFNyu3Xqx362M5WrHKnRP4A+vzB3YClKk=</latexit><latexit sha1_base64="P47SsFqzSSxa0RYDRpj/aiGYLv0=">AAAB+3icbVDLSsNAFL2pr1pftS7dDBbBVUlEsMuKG5cV7APaECaTSTt0MgkzE7GE/IobF4q49Ufc+TdO2iy09cAwh3PuZc4cP+FMadv+tiobm1vbO9Xd2t7+weFR/bjRV3EqCe2RmMdy6GNFORO0p5nmdJhIiiOf04E/uy38wSOVisXiQc8T6kZ4IljICNZG8uqNbOzHPFDzyFzZTZ57jldv2i17AbROnJI0oUTXq3+Ng5ikERWacKzUyLET7WZYakY4zWvjVNEEkxme0JGhAkdUudkie47OjRKgMJbmCI0W6u+NDEeqSGcmI6ynatUrxP+8UarDtpsxkaSaCrJ8KEw50jEqikABk5RoPjcEE8lMVkSmWGKiTV01U4Kz+uV10r9sOXbLub9qdtplHVU4hTO4AAeuoQN30IUeEHiCZ3iFNyu3Xqx362M5WrHKnRP4A+vzB3YClKk=</latexit><latexit sha1_base64="P47SsFqzSSxa0RYDRpj/aiGYLv0=">AAAB+3icbVDLSsNAFL2pr1pftS7dDBbBVUlEsMuKG5cV7APaECaTSTt0MgkzE7GE/IobF4q49Ufc+TdO2iy09cAwh3PuZc4cP+FMadv+tiobm1vbO9Xd2t7+weFR/bjRV3EqCe2RmMdy6GNFORO0p5nmdJhIiiOf04E/uy38wSOVisXiQc8T6kZ4IljICNZG8uqNbOzHPFDzyFzZTZ57jldv2i17AbROnJI0oUTXq3+Ng5ikERWacKzUyLET7WZYakY4zWvjVNEEkxme0JGhAkdUudkie47OjRKgMJbmCI0W6u+NDEeqSGcmI6ynatUrxP+8UarDtpsxkaSaCrJ8KEw50jEqikABk5RoPjcEE8lMVkSmWGKiTV01U4Kz+uV10r9sOXbLub9qdtplHVU4hTO4AAeuoQN30IUeEHiCZ3iFNyu3Xqx362M5WrHKnRP4A+vzB3YClKk=</latexit>

A2<latexit sha1_base64="zXIuX9UsBxGdVPiWOSYVO26NZBU=">AAAB+3icbVDLSsNAFL3xWeur1qWbwSK4KkkR7LLixmUF+4A2hMlk0g6dZMLMRCwhv+LGhSJu/RF3/o2TNgttPTDM4Zx7mTPHTzhT2ra/rY3Nre2d3cpedf/g8Oi4dlLvK5FKQntEcCGHPlaUs5j2NNOcDhNJceRzOvBnt4U/eKRSMRE/6HlC3QhPYhYygrWRvFo9G/uCB2oemSu7yXOv5dUadtNeAK0TpyQNKNH1al/jQJA0orEmHCs1cuxEuxmWmhFO8+o4VTTBZIYndGRojCOq3GyRPUcXRglQKKQ5sUYL9fdGhiNVpDOTEdZTteoV4n/eKNVh281YnKSaxmT5UJhypAUqikABk5RoPjcEE8lMVkSmWGKiTV1VU4Kz+uV10m81Hbvp3F81Ou2yjgqcwTlcggPX0IE76EIPCDzBM7zCm5VbL9a79bEc3bDKnVP4A+vzB3eGlKo=</latexit><latexit sha1_base64="zXIuX9UsBxGdVPiWOSYVO26NZBU=">AAAB+3icbVDLSsNAFL3xWeur1qWbwSK4KkkR7LLixmUF+4A2hMlk0g6dZMLMRCwhv+LGhSJu/RF3/o2TNgttPTDM4Zx7mTPHTzhT2ra/rY3Nre2d3cpedf/g8Oi4dlLvK5FKQntEcCGHPlaUs5j2NNOcDhNJceRzOvBnt4U/eKRSMRE/6HlC3QhPYhYygrWRvFo9G/uCB2oemSu7yXOv5dUadtNeAK0TpyQNKNH1al/jQJA0orEmHCs1cuxEuxmWmhFO8+o4VTTBZIYndGRojCOq3GyRPUcXRglQKKQ5sUYL9fdGhiNVpDOTEdZTteoV4n/eKNVh281YnKSaxmT5UJhypAUqikABk5RoPjcEE8lMVkSmWGKiTV1VU4Kz+uV10m81Hbvp3F81Ou2yjgqcwTlcggPX0IE76EIPCDzBM7zCm5VbL9a79bEc3bDKnVP4A+vzB3eGlKo=</latexit><latexit sha1_base64="zXIuX9UsBxGdVPiWOSYVO26NZBU=">AAAB+3icbVDLSsNAFL3xWeur1qWbwSK4KkkR7LLixmUF+4A2hMlk0g6dZMLMRCwhv+LGhSJu/RF3/o2TNgttPTDM4Zx7mTPHTzhT2ra/rY3Nre2d3cpedf/g8Oi4dlLvK5FKQntEcCGHPlaUs5j2NNOcDhNJceRzOvBnt4U/eKRSMRE/6HlC3QhPYhYygrWRvFo9G/uCB2oemSu7yXOv5dUadtNeAK0TpyQNKNH1al/jQJA0orEmHCs1cuxEuxmWmhFO8+o4VTTBZIYndGRojCOq3GyRPUcXRglQKKQ5sUYL9fdGhiNVpDOTEdZTteoV4n/eKNVh281YnKSaxmT5UJhypAUqikABk5RoPjcEE8lMVkSmWGKiTV1VU4Kz+uV10m81Hbvp3F81Ou2yjgqcwTlcggPX0IE76EIPCDzBM7zCm5VbL9a79bEc3bDKnVP4A+vzB3eGlKo=</latexit><latexit sha1_base64="zXIuX9UsBxGdVPiWOSYVO26NZBU=">AAAB+3icbVDLSsNAFL3xWeur1qWbwSK4KkkR7LLixmUF+4A2hMlk0g6dZMLMRCwhv+LGhSJu/RF3/o2TNgttPTDM4Zx7mTPHTzhT2ra/rY3Nre2d3cpedf/g8Oi4dlLvK5FKQntEcCGHPlaUs5j2NNOcDhNJceRzOvBnt4U/eKRSMRE/6HlC3QhPYhYygrWRvFo9G/uCB2oemSu7yXOv5dUadtNeAK0TpyQNKNH1al/jQJA0orEmHCs1cuxEuxmWmhFO8+o4VTTBZIYndGRojCOq3GyRPUcXRglQKKQ5sUYL9fdGhiNVpDOTEdZTteoV4n/eKNVh281YnKSaxmT5UJhypAUqikABk5RoPjcEE8lMVkSmWGKiTV1VU4Kz+uV10m81Hbvp3F81Ou2yjgqcwTlcggPX0IE76EIPCDzBM7zCm5VbL9a79bEc3bDKnVP4A+vzB3eGlKo=</latexit>

C<latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="feYzCSbqj61LrrcowietRi1DSP8=">AAAB5HicbVDLSgNBEOz1GddX9OplMAiewq4XPQpePEYwD0hCmJ3tTYbMzi4zvUJY8gMevIh49Zu8+TdOHqAmFjQUVd10d0W5kpaC4Mvb2Nza3tmt7Pn7B/7h0XH1pGWzwghsikxlphNxi0pqbJIkhZ3cIE8jhe1ofDfz209orMz0I01y7Kd8qGUiBScnNQbVWlAP5mDrJFySGiwxqH724kwUKWoSilvbDYOc+iU3JIXCqd8rLOZcjPkQu45qnqLtl/Mzp+zCKTFLMuNKE5urvydKnlo7SSPXmXIa2VVvJv7ndQtKbvql1HlBqMViUVIoRhmb/cxiaVCQmjjChZHuViZG3HBBLhnfZRCufrxOWlf1MKiHDz9hQAXO4BwuIYRruIV7aEATBMTwDK/eyHvx3rz3ReOGt5w4hT/wPr4BGDqLAg==</latexit><latexit sha1_base64="Ek295dnaWqDA7CVa33LsfvGS6DU=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKex6MUchF48RzAOSJcxOZpMxszPLTK8QlvyDFw+KePV/vPk3Th6gJhY0FFXddHdFqRQWff/LK2xsbm3vFHdLe/sHh0fl45OW1ZlhvMm01KYTUculULyJAiXvpIbTJJK8HY3rM7/9yI0VWt3jJOVhQodKxIJRdFKrx6gk9X654lf9Ocg6CZakAks0+uXP3kCzLOEKmaTWdgM/xTCnBgWTfFrqZZanlI3pkHcdVTThNszn107JhVMGJNbGlUIyV39P5DSxdpJErjOhOLKr3kz8z+tmGNfCXKg0Q67YYlGcSYKazF4nA2E4QzlxhDIj3K2EjaihDF1AJRdCsPryOmldVQO/Gtz9pAFFOINzuIQAruEGbqEBTWDwAE/wAq+e9p69N+990VrwljOn8Afexze0I44f</latexit><latexit sha1_base64="k/5lhehzUAL44uXOMT1KXOXhSyo=">AAAB7XicdVDLSgMxFL1TX7W+qi7dBIvgqkwqat0J3bisYB/QDiWTZtrYTDIkGaEM/Qc3LhRx6/+4829MH+IDPXDhcM693HtPmAhurO+/e7ml5ZXVtfx6YWNza3unuLvXNCrVlDWoEkq3Q2KY4JI1LLeCtRPNSBwK1gpHtanfumPacCVv7DhhQUwGkkecEuukZpcSgWq9YskvV6uVk9MT5Jf9GRzBPr44wwgvlBIsUO8V37p9RdOYSUsFMaaD/cQGGdGWU8EmhW5qWELoiAxYx1FJYmaCbHbtBB05pY8ipV1Ji2bq94mMxMaM49B1xsQOzW9vKv7ldVIbVYOMyyS1TNL5oigVyCo0fR31uWbUirEjhGrubkV0SDSh1gVUcCF8for+J82Ki6WMr7/SgDwcwCEcA4ZzuIQrqEMDKNzCPTzCk6e8B+/Ze5m35rzFzD78gPf6AfiPjk4=</latexit><latexit sha1_base64="VZr9pllCNJZ9jZxKm/RrdwKz36k=">AAAB7XicdVBNSwMxEJ2tX7V+VT16CRbBU8m2qOut0IvHCrYW2qVk02wbm80uSVYoS/+DFw+KePX/ePPfmLYrqOiDgcd7M8zMCxLBtcH4wymsrK6tbxQ3S1vbO7t75f2Djo5TRVmbxiJW3YBoJrhkbcONYN1EMRIFgt0Gk+bcv71nSvNY3phpwvyIjCQPOSXGSp0+JQI1B+UKrnperX5WR7iKF7DExe7luYvcXKlAjtag/N4fxjSNmDRUEK17Lk6MnxFlOBVsVuqnmiWETsiI9SyVJGLazxbXztCJVYYojJUtadBC/T6RkUjraRTYzoiYsf7tzcW/vF5qQs/PuExSwyRdLgpTgUyM5q+jIVeMGjG1hFDF7a2Ijoki1NiASjaEr0/R/6RTs7FU3WtcaXh5HEU4gmM4BRcuoAFX0II2ULiDB3iCZyd2Hp0X53XZWnDymUP4AeftExz2jsY=</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit><latexit sha1_base64="dAVmNH6maAgKZO8jWBlns74N3Ws=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoPgKewmPtZbIBePEcwDkiXMTmaTMbMzy8ysEEL+wYsHRbz6P978GyfJCipa0FBUddPdFSacaeO6H05uZXVtfSO/Wdja3tndK+4ftLRMFaFNIrlUnRBrypmgTcMMp51EURyHnLbDcX3ut++p0kyKWzNJaBDjoWARI9hYqdUjmKN6v1hyy75fqZ5XkVt2F7DEc72rCw95mVKCDI1+8b03kCSNqTCEY627npuYYIqVYYTTWaGXappgMsZD2rVU4JjqYLq4doZOrDJAkVS2hEEL9fvEFMdaT+LQdsbYjPRvby7+5XVTE/nBlIkkNVSQ5aIo5chINH8dDZiixPCJJZgoZm9FZIQVJsYGVLAhfH2K/ietio2l7N2clWp+FkcejuAYTsGDS6jBNTSgCQTu4AGe4NmRzqPz4rwuW3NONnMIP+C8fQIeNo7K</latexit>

RC<latexit sha1_base64="BEHK0GWawytbIpGjE0AzCNiD9ZI=">AAAB7nicdVDLSsNAFL2pr1pfVZduBovgqiStj7grdOOyin1AG8pkOmmHTiZhZiKU0I9w40IRt36PO//GaRpBRQ9cOJxzL/fe48ecKW3bH1ZhZXVtfaO4Wdra3tndK+8fdFSUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT5sLv3tPpWKRuNOzmHohHgsWMIK1kboDgjm6bQ7LFbvqurX6eR3ZVTuDIY7tXF04yMmVCuRoDcvvg1FEkpAKTThWqu/YsfZSLDUjnM5Lg0TRGJMpHtO+oQKHVHlpdu4cnRhlhIJImhIaZer3iRSHSs1C33SGWE/Ub28h/uX1Ex24XspEnGgqyHJRkHCkI7T4HY2YpETzmSGYSGZuRWSCJSbaJFQyIXx9iv4nnZqJpercnFUabh5HEY7gGE7BgUtowDW0oA0EpvAAT/Bsxdaj9WK9LlsLVj5zCD9gvX0CwWqPJg==</latexit><latexit sha1_base64="BEHK0GWawytbIpGjE0AzCNiD9ZI=">AAAB7nicdVDLSsNAFL2pr1pfVZduBovgqiStj7grdOOyin1AG8pkOmmHTiZhZiKU0I9w40IRt36PO//GaRpBRQ9cOJxzL/fe48ecKW3bH1ZhZXVtfaO4Wdra3tndK+8fdFSUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT5sLv3tPpWKRuNOzmHohHgsWMIK1kboDgjm6bQ7LFbvqurX6eR3ZVTuDIY7tXF04yMmVCuRoDcvvg1FEkpAKTThWqu/YsfZSLDUjnM5Lg0TRGJMpHtO+oQKHVHlpdu4cnRhlhIJImhIaZer3iRSHSs1C33SGWE/Ub28h/uX1Ex24XspEnGgqyHJRkHCkI7T4HY2YpETzmSGYSGZuRWSCJSbaJFQyIXx9iv4nnZqJpercnFUabh5HEY7gGE7BgUtowDW0oA0EpvAAT/Bsxdaj9WK9LlsLVj5zCD9gvX0CwWqPJg==</latexit><latexit sha1_base64="BEHK0GWawytbIpGjE0AzCNiD9ZI=">AAAB7nicdVDLSsNAFL2pr1pfVZduBovgqiStj7grdOOyin1AG8pkOmmHTiZhZiKU0I9w40IRt36PO//GaRpBRQ9cOJxzL/fe48ecKW3bH1ZhZXVtfaO4Wdra3tndK+8fdFSUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT5sLv3tPpWKRuNOzmHohHgsWMIK1kboDgjm6bQ7LFbvqurX6eR3ZVTuDIY7tXF04yMmVCuRoDcvvg1FEkpAKTThWqu/YsfZSLDUjnM5Lg0TRGJMpHtO+oQKHVHlpdu4cnRhlhIJImhIaZer3iRSHSs1C33SGWE/Ub28h/uX1Ex24XspEnGgqyHJRkHCkI7T4HY2YpETzmSGYSGZuRWSCJSbaJFQyIXx9iv4nnZqJpercnFUabh5HEY7gGE7BgUtowDW0oA0EpvAAT/Bsxdaj9WK9LlsLVj5zCD9gvX0CwWqPJg==</latexit><latexit sha1_base64="BEHK0GWawytbIpGjE0AzCNiD9ZI=">AAAB7nicdVDLSsNAFL2pr1pfVZduBovgqiStj7grdOOyin1AG8pkOmmHTiZhZiKU0I9w40IRt36PO//GaRpBRQ9cOJxzL/fe48ecKW3bH1ZhZXVtfaO4Wdra3tndK+8fdFSUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT5sLv3tPpWKRuNOzmHohHgsWMIK1kboDgjm6bQ7LFbvqurX6eR3ZVTuDIY7tXF04yMmVCuRoDcvvg1FEkpAKTThWqu/YsfZSLDUjnM5Lg0TRGJMpHtO+oQKHVHlpdu4cnRhlhIJImhIaZer3iRSHSs1C33SGWE/Ub28h/uX1Ex24XspEnGgqyHJRkHCkI7T4HY2YpETzmSGYSGZuRWSCJSbaJFQyIXx9iv4nnZqJpercnFUabh5HEY7gGE7BgUtowDW0oA0EpvAAT/Bsxdaj9WK9LlsLVj5zCD9gvX0CwWqPJg==</latexit>

real space<latexit sha1_base64="mVnO8BgKIMYKFgLFQJk55y1mmQo=">AAAB+nicbVDLSgMxFM34rPU11aWbYBFclRkR7LLgxmUF+4C2lEx6pw3NY0gyahn7KW5cKOLWL3Hn35i2s9DWA4HDOfckNydKODM2CL69tfWNza3twk5xd2//4NAvHTWNSjWFBlVc6XZEDHAmoWGZ5dBONBARcWhF4+uZ37oHbZiSd3aSQE+QoWQxo8Q6qe+XuiJSj5mLcGwSQmHa98tBJZgDr5IwJ2WUo973v7oDRVMB0lJOjOmEQWJ7GdGWUQ7TYjc14G4ekyF0HJVEgOll89Wn+MwpAxwr7Y60eK7+TmREGDMRkZsUxI7MsjcT//M6qY2rvYzJJLUg6eKhOOXYKjzrAQ+YBmr5xBFCNXO7YjoimlDr2iq6EsLlL6+S5kUlDCrh7WW5Vs3rKKATdIrOUYiuUA3doDpqIIoe0DN6RW/ek/fivXsfi9E1L88coz/wPn8AjrqUJQ==</latexit><latexit sha1_base64="mVnO8BgKIMYKFgLFQJk55y1mmQo=">AAAB+nicbVDLSgMxFM34rPU11aWbYBFclRkR7LLgxmUF+4C2lEx6pw3NY0gyahn7KW5cKOLWL3Hn35i2s9DWA4HDOfckNydKODM2CL69tfWNza3twk5xd2//4NAvHTWNSjWFBlVc6XZEDHAmoWGZ5dBONBARcWhF4+uZ37oHbZiSd3aSQE+QoWQxo8Q6qe+XuiJSj5mLcGwSQmHa98tBJZgDr5IwJ2WUo973v7oDRVMB0lJOjOmEQWJ7GdGWUQ7TYjc14G4ekyF0HJVEgOll89Wn+MwpAxwr7Y60eK7+TmREGDMRkZsUxI7MsjcT//M6qY2rvYzJJLUg6eKhOOXYKjzrAQ+YBmr5xBFCNXO7YjoimlDr2iq6EsLlL6+S5kUlDCrh7WW5Vs3rKKATdIrOUYiuUA3doDpqIIoe0DN6RW/ek/fivXsfi9E1L88coz/wPn8AjrqUJQ==</latexit><latexit sha1_base64="mVnO8BgKIMYKFgLFQJk55y1mmQo=">AAAB+nicbVDLSgMxFM34rPU11aWbYBFclRkR7LLgxmUF+4C2lEx6pw3NY0gyahn7KW5cKOLWL3Hn35i2s9DWA4HDOfckNydKODM2CL69tfWNza3twk5xd2//4NAvHTWNSjWFBlVc6XZEDHAmoWGZ5dBONBARcWhF4+uZ37oHbZiSd3aSQE+QoWQxo8Q6qe+XuiJSj5mLcGwSQmHa98tBJZgDr5IwJ2WUo973v7oDRVMB0lJOjOmEQWJ7GdGWUQ7TYjc14G4ekyF0HJVEgOll89Wn+MwpAxwr7Y60eK7+TmREGDMRkZsUxI7MsjcT//M6qY2rvYzJJLUg6eKhOOXYKjzrAQ+YBmr5xBFCNXO7YjoimlDr2iq6EsLlL6+S5kUlDCrh7WW5Vs3rKKATdIrOUYiuUA3doDpqIIoe0DN6RW/ek/fivXsfi9E1L88coz/wPn8AjrqUJQ==</latexit><latexit sha1_base64="mVnO8BgKIMYKFgLFQJk55y1mmQo=">AAAB+nicbVDLSgMxFM34rPU11aWbYBFclRkR7LLgxmUF+4C2lEx6pw3NY0gyahn7KW5cKOLWL3Hn35i2s9DWA4HDOfckNydKODM2CL69tfWNza3twk5xd2//4NAvHTWNSjWFBlVc6XZEDHAmoWGZ5dBONBARcWhF4+uZ37oHbZiSd3aSQE+QoWQxo8Q6qe+XuiJSj5mLcGwSQmHa98tBJZgDr5IwJ2WUo973v7oDRVMB0lJOjOmEQWJ7GdGWUQ7TYjc14G4ekyF0HJVEgOll89Wn+MwpAxwr7Y60eK7+TmREGDMRkZsUxI7MsjcT//M6qY2rvYzJJLUg6eKhOOXYKjzrAQ+YBmr5xBFCNXO7YjoimlDr2iq6EsLlL6+S5kUlDCrh7WW5Vs3rKKATdIrOUYiuUA3doDpqIIoe0DN6RW/ek/fivXsfi9E1L88coz/wPn8AjrqUJQ==</latexit>

reciprocal space

<latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit><latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit><latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit><latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit>

discrete Fourier series

f(x) = f(x+R) ) f(x) =X

G

f

G

e

iGx

, f

G

=1

VC

Z

Cd

Dx f(x)e�iGx

<latexit sha1_base64="GoO67tOA3p3jse6njqpTPhq3t3I=">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</latexit><latexit sha1_base64="GoO67tOA3p3jse6njqpTPhq3t3I=">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</latexit><latexit sha1_base64="GoO67tOA3p3jse6njqpTPhq3t3I=">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</latexit><latexit sha1_base64="GoO67tOA3p3jse6njqpTPhq3t3I=">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</latexit>










reciprocal space


periodic boundaries

L = L1 ⇥ · · ·⇥ LD<latexit sha1_base64="kUaSq1yjFEH8XDsH+8jdw/Txwps=">AAACBnicbZDLSsNAFIYn9VbrLepShMEiuCpJEXQjFHThoosK9gJNCJPJpB06uTBzIpTQlRtfxY0LRdz6DO58G6dtFtr6w8DHf87hzPn9VHAFlvVtlFZW19Y3ypuVre2d3T1z/6CjkkxS1qaJSGTPJ4oJHrM2cBCsl0pGIl+wrj+6nta7D0wqnsT3ME6ZG5FBzENOCWjLM4+b+Ao3PdsBHjHl0CABNWft3nhm1apZM+FlsAuookItz/xygoRmEYuBCqJU37ZScHMigVPBJhUnUywldEQGrK8xJnqRm8/OmOBT7QQ4TKR+MeCZ+3siJ5FS48jXnRGBoVqsTc3/av0Mwks353GaAYvpfFGYCQwJnmaCAy4ZBTHWQKjk+q+YDokkFHRyFR2CvXjyMnTqNduq2Xfn1Ua9iKOMjtAJOkM2ukANdItaqI0oekTP6BW9GU/Gi/FufMxbS0Yxc4j+yPj8AZ7wl+Y=</latexit><latexit sha1_base64="kUaSq1yjFEH8XDsH+8jdw/Txwps=">AAACBnicbZDLSsNAFIYn9VbrLepShMEiuCpJEXQjFHThoosK9gJNCJPJpB06uTBzIpTQlRtfxY0LRdz6DO58G6dtFtr6w8DHf87hzPn9VHAFlvVtlFZW19Y3ypuVre2d3T1z/6CjkkxS1qaJSGTPJ4oJHrM2cBCsl0pGIl+wrj+6nta7D0wqnsT3ME6ZG5FBzENOCWjLM4+b+Ao3PdsBHjHl0CABNWft3nhm1apZM+FlsAuookItz/xygoRmEYuBCqJU37ZScHMigVPBJhUnUywldEQGrK8xJnqRm8/OmOBT7QQ4TKR+MeCZ+3siJ5FS48jXnRGBoVqsTc3/av0Mwks353GaAYvpfFGYCQwJnmaCAy4ZBTHWQKjk+q+YDokkFHRyFR2CvXjyMnTqNduq2Xfn1Ua9iKOMjtAJOkM2ukANdItaqI0oekTP6BW9GU/Gi/FufMxbS0Yxc4j+yPj8AZ7wl+Y=</latexit><latexit sha1_base64="kUaSq1yjFEH8XDsH+8jdw/Txwps=">AAACBnicbZDLSsNAFIYn9VbrLepShMEiuCpJEXQjFHThoosK9gJNCJPJpB06uTBzIpTQlRtfxY0LRdz6DO58G6dtFtr6w8DHf87hzPn9VHAFlvVtlFZW19Y3ypuVre2d3T1z/6CjkkxS1qaJSGTPJ4oJHrM2cBCsl0pGIl+wrj+6nta7D0wqnsT3ME6ZG5FBzENOCWjLM4+b+Ao3PdsBHjHl0CABNWft3nhm1apZM+FlsAuookItz/xygoRmEYuBCqJU37ZScHMigVPBJhUnUywldEQGrK8xJnqRm8/OmOBT7QQ4TKR+MeCZ+3siJ5FS48jXnRGBoVqsTc3/av0Mwks353GaAYvpfFGYCQwJnmaCAy4ZBTHWQKjk+q+YDokkFHRyFR2CvXjyMnTqNduq2Xfn1Ua9iKOMjtAJOkM2ukANdItaqI0oekTP6BW9GU/Gi/FufMxbS0Yxc4j+yPj8AZ7wl+Y=</latexit><latexit sha1_base64="kUaSq1yjFEH8XDsH+8jdw/Txwps=">AAACBnicbZDLSsNAFIYn9VbrLepShMEiuCpJEXQjFHThoosK9gJNCJPJpB06uTBzIpTQlRtfxY0LRdz6DO58G6dtFtr6w8DHf87hzPn9VHAFlvVtlFZW19Y3ypuVre2d3T1z/6CjkkxS1qaJSGTPJ4oJHrM2cBCsl0pGIl+wrj+6nta7D0wqnsT3ME6ZG5FBzENOCWjLM4+b+Ao3PdsBHjHl0CABNWft3nhm1apZM+FlsAuookItz/xygoRmEYuBCqJU37ZScHMigVPBJhUnUywldEQGrK8xJnqRm8/OmOBT7QQ4TKR+MeCZ+3siJ5FS48jXnRGBoVqsTc3/av0Mwks353GaAYvpfFGYCQwJnmaCAy4ZBTHWQKjk+q+YDokkFHRyFR2CvXjyMnTqNduq2Xfn1Ua9iKOMjtAJOkM2ukANdItaqI0oekTP6BW9GU/Gi/FufMxbS0Yxc4j+yPj8AZ7wl+Y=</latexit>

V = det(A1, ...,AD)<latexit sha1_base64="oA6qQkWpbEg2adrHXy8BRTpLwhg=">AAACG3icbVDLSsNAFJ3UV62vqEs3g0WoUEJSBN0IFV24rGAf0IQwmUzboZNJmJkIJfQ/3Pgrblwo4kpw4d84abPQ1gPDHM65l3vvCRJGpbLtb6O0srq2vlHerGxt7+zumfsHHRmnApM2jlksegGShFFO2ooqRnqJICgKGOkG4+vc7z4QIWnM79UkIV6EhpwOKEZKS77Z6Fy6IVGwlrlBzEI5ifQHr6a+U4eWZdXhon5z6ptV27JngMvEKUgVFGj55qcbxjiNCFeYISn7jp0oL0NCUczItOKmkiQIj9GQ9DXlKCLSy2a3TeGJVkI4iIV+XMGZ+rsjQ5HMt9OVEVIjuejl4n9eP1WDCy+jPEkV4Xg+aJAyqGKYBwVDKghWbKIJwoLqXSEeIYGw0nFWdAjO4snLpNOwHNty7s6qzUYRRxkcgWNQAw44B01wC1qgDTB4BM/gFbwZT8aL8W58zEtLRtFzCP7A+PoB1IWfWQ==</latexit><latexit sha1_base64="oA6qQkWpbEg2adrHXy8BRTpLwhg=">AAACG3icbVDLSsNAFJ3UV62vqEs3g0WoUEJSBN0IFV24rGAf0IQwmUzboZNJmJkIJfQ/3Pgrblwo4kpw4d84abPQ1gPDHM65l3vvCRJGpbLtb6O0srq2vlHerGxt7+zumfsHHRmnApM2jlksegGShFFO2ooqRnqJICgKGOkG4+vc7z4QIWnM79UkIV6EhpwOKEZKS77Z6Fy6IVGwlrlBzEI5ifQHr6a+U4eWZdXhon5z6ptV27JngMvEKUgVFGj55qcbxjiNCFeYISn7jp0oL0NCUczItOKmkiQIj9GQ9DXlKCLSy2a3TeGJVkI4iIV+XMGZ+rsjQ5HMt9OVEVIjuejl4n9eP1WDCy+jPEkV4Xg+aJAyqGKYBwVDKghWbKIJwoLqXSEeIYGw0nFWdAjO4snLpNOwHNty7s6qzUYRRxkcgWNQAw44B01wC1qgDTB4BM/gFbwZT8aL8W58zEtLRtFzCP7A+PoB1IWfWQ==</latexit><latexit sha1_base64="oA6qQkWpbEg2adrHXy8BRTpLwhg=">AAACG3icbVDLSsNAFJ3UV62vqEs3g0WoUEJSBN0IFV24rGAf0IQwmUzboZNJmJkIJfQ/3Pgrblwo4kpw4d84abPQ1gPDHM65l3vvCRJGpbLtb6O0srq2vlHerGxt7+zumfsHHRmnApM2jlksegGShFFO2ooqRnqJICgKGOkG4+vc7z4QIWnM79UkIV6EhpwOKEZKS77Z6Fy6IVGwlrlBzEI5ifQHr6a+U4eWZdXhon5z6ptV27JngMvEKUgVFGj55qcbxjiNCFeYISn7jp0oL0NCUczItOKmkiQIj9GQ9DXlKCLSy2a3TeGJVkI4iIV+XMGZ+rsjQ5HMt9OVEVIjuejl4n9eP1WDCy+jPEkV4Xg+aJAyqGKYBwVDKghWbKIJwoLqXSEeIYGw0nFWdAjO4snLpNOwHNty7s6qzUYRRxkcgWNQAw44B01wC1qgDTB4BM/gFbwZT8aL8W58zEtLRtFzCP7A+PoB1IWfWQ==</latexit><latexit sha1_base64="oA6qQkWpbEg2adrHXy8BRTpLwhg=">AAACG3icbVDLSsNAFJ3UV62vqEs3g0WoUEJSBN0IFV24rGAf0IQwmUzboZNJmJkIJfQ/3Pgrblwo4kpw4d84abPQ1gPDHM65l3vvCRJGpbLtb6O0srq2vlHerGxt7+zumfsHHRmnApM2jlksegGShFFO2ooqRnqJICgKGOkG4+vc7z4QIWnM79UkIV6EhpwOKEZKS77Z6Fy6IVGwlrlBzEI5ifQHr6a+U4eWZdXhon5z6ptV27JngMvEKUgVFGj55qcbxjiNCFeYISn7jp0oL0NCUczItOKmkiQIj9GQ9DXlKCLSy2a3TeGJVkI4iIV+XMGZ+rsjQ5HMt9OVEVIjuejl4n9eP1WDCy+jPEkV4Xg+aJAyqGKYBwVDKghWbKIJwoLqXSEeIYGw0nFWdAjO4snLpNOwHNty7s6qzUYRRxkcgWNQAw44B01wC1qgDTB4BM/gFbwZT8aL8W58zEtLRtFzCP7A+PoB1IWfWQ==</latexit>

x ⌘ x+A↵<latexit sha1_base64="nC8j5UUbeFlL3qUWE7nD9dV+XfA=">AAACJ3icbVDLSsNAFJ3UV62vqEs3g0UQhJIUQVdSceOygn1AE8JkMm2HTjJxZlIsIX/jxl9xI6iILv0TJ20WtvXAMIdz7uXee/yYUaks69sorayurW+UNytb2zu7e+b+QVvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rJ/c6YCEl5dK8mMXFDNIhon2KktOSZV6njcxbISag/+Jg55CGhY7igwrN55TrzHMTiIfLMqlWzpoDLxC5IFRRoeuabE3CchCRSmCEpe7YVKzdFQlHMSFZxEklihEdoQHqaRigk0k2nd2bwRCsB7HOhX6TgVP3bkaJQ5hvqyhCpoVz0cvE/r5eo/qWb0ihOFInwbFA/YVBxmIcGAyoIVmyiCcKC6l0hHiKBsNLRVnQI9uLJy6Rdr9lWzb47rzbqRRxlcASOwSmwwQVogFvQBC2AwRN4Ae/gw3g2Xo1P42tWWjKKnkMwB+PnF9R1pws=</latexit><latexit sha1_base64="nC8j5UUbeFlL3qUWE7nD9dV+XfA=">AAACJ3icbVDLSsNAFJ3UV62vqEs3g0UQhJIUQVdSceOygn1AE8JkMm2HTjJxZlIsIX/jxl9xI6iILv0TJ20WtvXAMIdz7uXee/yYUaks69sorayurW+UNytb2zu7e+b+QVvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rJ/c6YCEl5dK8mMXFDNIhon2KktOSZV6njcxbISag/+Jg55CGhY7igwrN55TrzHMTiIfLMqlWzpoDLxC5IFRRoeuabE3CchCRSmCEpe7YVKzdFQlHMSFZxEklihEdoQHqaRigk0k2nd2bwRCsB7HOhX6TgVP3bkaJQ5hvqyhCpoVz0cvE/r5eo/qWb0ihOFInwbFA/YVBxmIcGAyoIVmyiCcKC6l0hHiKBsNLRVnQI9uLJy6Rdr9lWzb47rzbqRRxlcASOwSmwwQVogFvQBC2AwRN4Ae/gw3g2Xo1P42tWWjKKnkMwB+PnF9R1pws=</latexit><latexit sha1_base64="nC8j5UUbeFlL3qUWE7nD9dV+XfA=">AAACJ3icbVDLSsNAFJ3UV62vqEs3g0UQhJIUQVdSceOygn1AE8JkMm2HTjJxZlIsIX/jxl9xI6iILv0TJ20WtvXAMIdz7uXee/yYUaks69sorayurW+UNytb2zu7e+b+QVvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rJ/c6YCEl5dK8mMXFDNIhon2KktOSZV6njcxbISag/+Jg55CGhY7igwrN55TrzHMTiIfLMqlWzpoDLxC5IFRRoeuabE3CchCRSmCEpe7YVKzdFQlHMSFZxEklihEdoQHqaRigk0k2nd2bwRCsB7HOhX6TgVP3bkaJQ5hvqyhCpoVz0cvE/r5eo/qWb0ihOFInwbFA/YVBxmIcGAyoIVmyiCcKC6l0hHiKBsNLRVnQI9uLJy6Rdr9lWzb47rzbqRRxlcASOwSmwwQVogFvQBC2AwRN4Ae/gw3g2Xo1P42tWWjKKnkMwB+PnF9R1pws=</latexit><latexit sha1_base64="nC8j5UUbeFlL3qUWE7nD9dV+XfA=">AAACJ3icbVDLSsNAFJ3UV62vqEs3g0UQhJIUQVdSceOygn1AE8JkMm2HTjJxZlIsIX/jxl9xI6iILv0TJ20WtvXAMIdz7uXee/yYUaks69sorayurW+UNytb2zu7e+b+QVvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rJ/c6YCEl5dK8mMXFDNIhon2KktOSZV6njcxbISag/+Jg55CGhY7igwrN55TrzHMTiIfLMqlWzpoDLxC5IFRRoeuabE3CchCRSmCEpe7YVKzdFQlHMSFZxEklihEdoQHqaRigk0k2nd2bwRCsB7HOhX6TgVP3bkaJQ5hvqyhCpoVz0cvE/r5eo/qWb0ihOFInwbFA/YVBxmIcGAyoIVmyiCcKC6l0hHiKBsNLRVnQI9uLJy6Rdr9lWzb47rzbqRRxlcASOwSmwwQVogFvQBC2AwRN4Ae/gw3g2Xo1P42tWWjKKnkMwB+PnF9R1pws=</latexit> eikA↵ = 1

<latexit sha1_base64="ZwLWrq/dYj3aGW8idKjTzwWbXAg=">AAACGHicbVC7TsMwFHXKq5RXgJHFokJiKkmFBAtSEQtjkehDakLkOE5r1XEi20GqonwGC7/CwgBCrN34G5w2A205kuXjc+6V7z1+wqhUlvVjVNbWNza3qtu1nd29/QPz8Kgr41Rg0sExi0XfR5IwyklHUcVIPxEERT4jPX98V/i9ZyIkjfmjmiTEjdCQ05BipLTkmRfkKaMwc/yYBXIS6QuO84Xnbe45iCUjlMMbaHtm3WpYM8BVYpekDkq0PXPqBDFOI8IVZkjKgW0lys2QUBQzktecVJIE4TEakoGmHEVEutlssRyeaSWAYSz04QrO1L8dGYpkMaaujJAayWWvEP/zBqkKr92M8iRVhOP5R2HKoIphkRIMqCBYsYkmCAuqZ4V4hATCSmdZ0yHYyyuvkm6zYVsN++Gy3mqWcVTBCTgF58AGV6AF7kEbdAAGL+ANfIBP49V4N76M73lpxSh7jsECjOkv0lOgJg==</latexit><latexit sha1_base64="ZwLWrq/dYj3aGW8idKjTzwWbXAg=">AAACGHicbVC7TsMwFHXKq5RXgJHFokJiKkmFBAtSEQtjkehDakLkOE5r1XEi20GqonwGC7/CwgBCrN34G5w2A205kuXjc+6V7z1+wqhUlvVjVNbWNza3qtu1nd29/QPz8Kgr41Rg0sExi0XfR5IwyklHUcVIPxEERT4jPX98V/i9ZyIkjfmjmiTEjdCQ05BipLTkmRfkKaMwc/yYBXIS6QuO84Xnbe45iCUjlMMbaHtm3WpYM8BVYpekDkq0PXPqBDFOI8IVZkjKgW0lys2QUBQzktecVJIE4TEakoGmHEVEutlssRyeaSWAYSz04QrO1L8dGYpkMaaujJAayWWvEP/zBqkKr92M8iRVhOP5R2HKoIphkRIMqCBYsYkmCAuqZ4V4hATCSmdZ0yHYyyuvkm6zYVsN++Gy3mqWcVTBCTgF58AGV6AF7kEbdAAGL+ANfIBP49V4N76M73lpxSh7jsECjOkv0lOgJg==</latexit><latexit sha1_base64="ZwLWrq/dYj3aGW8idKjTzwWbXAg=">AAACGHicbVC7TsMwFHXKq5RXgJHFokJiKkmFBAtSEQtjkehDakLkOE5r1XEi20GqonwGC7/CwgBCrN34G5w2A205kuXjc+6V7z1+wqhUlvVjVNbWNza3qtu1nd29/QPz8Kgr41Rg0sExi0XfR5IwyklHUcVIPxEERT4jPX98V/i9ZyIkjfmjmiTEjdCQ05BipLTkmRfkKaMwc/yYBXIS6QuO84Xnbe45iCUjlMMbaHtm3WpYM8BVYpekDkq0PXPqBDFOI8IVZkjKgW0lys2QUBQzktecVJIE4TEakoGmHEVEutlssRyeaSWAYSz04QrO1L8dGYpkMaaujJAayWWvEP/zBqkKr92M8iRVhOP5R2HKoIphkRIMqCBYsYkmCAuqZ4V4hATCSmdZ0yHYyyuvkm6zYVsN++Gy3mqWcVTBCTgF58AGV6AF7kEbdAAGL+ANfIBP49V4N76M73lpxSh7jsECjOkv0lOgJg==</latexit><latexit sha1_base64="ZwLWrq/dYj3aGW8idKjTzwWbXAg=">AAACGHicbVC7TsMwFHXKq5RXgJHFokJiKkmFBAtSEQtjkehDakLkOE5r1XEi20GqonwGC7/CwgBCrN34G5w2A205kuXjc+6V7z1+wqhUlvVjVNbWNza3qtu1nd29/QPz8Kgr41Rg0sExi0XfR5IwyklHUcVIPxEERT4jPX98V/i9ZyIkjfmjmiTEjdCQ05BipLTkmRfkKaMwc/yYBXIS6QuO84Xnbe45iCUjlMMbaHtm3WpYM8BVYpekDkq0PXPqBDFOI8IVZkjKgW0lys2QUBQzktecVJIE4TEakoGmHEVEutlssRyeaSWAYSz04QrO1L8dGYpkMaaujJAayWWvEP/zBqkKr92M8iRVhOP5R2HKoIphkRIMqCBYsYkmCAuqZ4V4hATCSmdZ0yHYyyuvkm6zYVsN++Gy3mqWcVTBCTgF58AGV6AF7kEbdAAGL+ANfIBP49V4N76M73lpxSh7jsECjOkv0lOgJg==</latexit>

V �Dk = (2⇡)D<latexit sha1_base64="KHlLDaDtPI9dIm3ZY3X5bmnCsgs=">AAACAnicbVDLSgNBEJyNrxhfq57Ey2AQIkjYDYJehIA5eIxgHpDdhNlJbzJk9sHMrBCW4MVf8eJBEa9+hTf/xkmyB00saCiquunu8mLOpLKsbyO3srq2vpHfLGxt7+zumfsHTRklgkKDRjwSbY9I4CyEhmKKQzsWQAKPQ8sb3Uz91gMIyaLwXo1jcAMyCJnPKFFa6plHTefcqQFXpFvDI4yvcanixOysW+uZRatszYCXiZ2RIspQ75lfTj+iSQChopxI2bGtWLkpEYpRDpOCk0iICR2RAXQ0DUkA0k1nL0zwqVb62I+ErlDhmfp7IiWBlOPA050BUUO56E3F/7xOovwrN2VhnCgI6XyRn3CsIjzNA/eZAKr4WBNCBdO3YjokglClUyvoEOzFl5dJs1K2rbJ9d1GsVrI48ugYnaASstElqqJbVEcNRNEjekav6M14Ml6Md+Nj3pozsplD9AfG5w+BLpTd</latexit><latexit sha1_base64="KHlLDaDtPI9dIm3ZY3X5bmnCsgs=">AAACAnicbVDLSgNBEJyNrxhfq57Ey2AQIkjYDYJehIA5eIxgHpDdhNlJbzJk9sHMrBCW4MVf8eJBEa9+hTf/xkmyB00saCiquunu8mLOpLKsbyO3srq2vpHfLGxt7+zumfsHTRklgkKDRjwSbY9I4CyEhmKKQzsWQAKPQ8sb3Uz91gMIyaLwXo1jcAMyCJnPKFFa6plHTefcqQFXpFvDI4yvcanixOysW+uZRatszYCXiZ2RIspQ75lfTj+iSQChopxI2bGtWLkpEYpRDpOCk0iICR2RAXQ0DUkA0k1nL0zwqVb62I+ErlDhmfp7IiWBlOPA050BUUO56E3F/7xOovwrN2VhnCgI6XyRn3CsIjzNA/eZAKr4WBNCBdO3YjokglClUyvoEOzFl5dJs1K2rbJ9d1GsVrI48ugYnaASstElqqJbVEcNRNEjekav6M14Ml6Md+Nj3pozsplD9AfG5w+BLpTd</latexit><latexit sha1_base64="KHlLDaDtPI9dIm3ZY3X5bmnCsgs=">AAACAnicbVDLSgNBEJyNrxhfq57Ey2AQIkjYDYJehIA5eIxgHpDdhNlJbzJk9sHMrBCW4MVf8eJBEa9+hTf/xkmyB00saCiquunu8mLOpLKsbyO3srq2vpHfLGxt7+zumfsHTRklgkKDRjwSbY9I4CyEhmKKQzsWQAKPQ8sb3Uz91gMIyaLwXo1jcAMyCJnPKFFa6plHTefcqQFXpFvDI4yvcanixOysW+uZRatszYCXiZ2RIspQ75lfTj+iSQChopxI2bGtWLkpEYpRDpOCk0iICR2RAXQ0DUkA0k1nL0zwqVb62I+ErlDhmfp7IiWBlOPA050BUUO56E3F/7xOovwrN2VhnCgI6XyRn3CsIjzNA/eZAKr4WBNCBdO3YjokglClUyvoEOzFl5dJs1K2rbJ9d1GsVrI48ugYnaASstElqqJbVEcNRNEjekav6M14Ml6Md+Nj3pozsplD9AfG5w+BLpTd</latexit><latexit sha1_base64="KHlLDaDtPI9dIm3ZY3X5bmnCsgs=">AAACAnicbVDLSgNBEJyNrxhfq57Ey2AQIkjYDYJehIA5eIxgHpDdhNlJbzJk9sHMrBCW4MVf8eJBEa9+hTf/xkmyB00saCiquunu8mLOpLKsbyO3srq2vpHfLGxt7+zumfsHTRklgkKDRjwSbY9I4CyEhmKKQzsWQAKPQ8sb3Uz91gMIyaLwXo1jcAMyCJnPKFFa6plHTefcqQFXpFvDI4yvcanixOysW+uZRatszYCXiZ2RIspQ75lfTj+iSQChopxI2bGtWLkpEYpRDpOCk0iICR2RAXQ0DUkA0k1nL0zwqVb62I+ErlDhmfp7IiWBlOPA050BUUO56E3F/7xOovwrN2VhnCgI6XyRn3CsIjzNA/eZAKr4WBNCBdO3YjokglClUyvoEOzFl5dJs1K2rbJ9d1GsVrI48ugYnaASstElqqJbVEcNRNEjekav6M14Ml6Md+Nj3pozsplD9AfG5w+BLpTd</latexit>

L = VRC/�Dk

<latexit sha1_base64="JZCaNuWBzvB4Fhg6CQVX04IWYps=">AAACBXicbVC7SgNBFL0bXzG+Vi21GAyCVdwNgjZCICksLKKYB2TXZXYySYbMPpiZFcKSxsZfsbFQxNZ/sPNvnCRbaOKBC4dz7uXee/yYM6ks69vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaPJeUspA3FFKftWFAc+Jy2/GF14rceqJAsCu/UKKZugPsh6zGClZY88/AaXaKmlzoEc3RbHaNT5NQoV/i+hoaeWbRK1hRokdgZKUKGumd+Od2IJAENFeFYyo5txcpNsVCMcDouOImkMSZD3KcdTUMcUOmm0y/G6FgrXdSLhK5Qoan6eyLFgZSjwNedAVYDOe9NxP+8TqJ6F27KwjhRNCSzRb2EIxWhSSSoywQlio80wUQwfSsiAywwUTq4gg7Bnn95kTTLJdsq2TdnxUo5iyMPB3AEJ2DDOVTgCurQAAKP8Ayv8GY8GS/Gu/Exa80Z2cw+/IHx+QMFD5ZM</latexit><latexit sha1_base64="JZCaNuWBzvB4Fhg6CQVX04IWYps=">AAACBXicbVC7SgNBFL0bXzG+Vi21GAyCVdwNgjZCICksLKKYB2TXZXYySYbMPpiZFcKSxsZfsbFQxNZ/sPNvnCRbaOKBC4dz7uXee/yYM6ks69vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaPJeUspA3FFKftWFAc+Jy2/GF14rceqJAsCu/UKKZugPsh6zGClZY88/AaXaKmlzoEc3RbHaNT5NQoV/i+hoaeWbRK1hRokdgZKUKGumd+Od2IJAENFeFYyo5txcpNsVCMcDouOImkMSZD3KcdTUMcUOmm0y/G6FgrXdSLhK5Qoan6eyLFgZSjwNedAVYDOe9NxP+8TqJ6F27KwjhRNCSzRb2EIxWhSSSoywQlio80wUQwfSsiAywwUTq4gg7Bnn95kTTLJdsq2TdnxUo5iyMPB3AEJ2DDOVTgCurQAAKP8Ayv8GY8GS/Gu/Exa80Z2cw+/IHx+QMFD5ZM</latexit><latexit sha1_base64="JZCaNuWBzvB4Fhg6CQVX04IWYps=">AAACBXicbVC7SgNBFL0bXzG+Vi21GAyCVdwNgjZCICksLKKYB2TXZXYySYbMPpiZFcKSxsZfsbFQxNZ/sPNvnCRbaOKBC4dz7uXee/yYM6ks69vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaPJeUspA3FFKftWFAc+Jy2/GF14rceqJAsCu/UKKZugPsh6zGClZY88/AaXaKmlzoEc3RbHaNT5NQoV/i+hoaeWbRK1hRokdgZKUKGumd+Od2IJAENFeFYyo5txcpNsVCMcDouOImkMSZD3KcdTUMcUOmm0y/G6FgrXdSLhK5Qoan6eyLFgZSjwNedAVYDOe9NxP+8TqJ6F27KwjhRNCSzRb2EIxWhSSSoywQlio80wUQwfSsiAywwUTq4gg7Bnn95kTTLJdsq2TdnxUo5iyMPB3AEJ2DDOVTgCurQAAKP8Ayv8GY8GS/Gu/Exa80Z2cw+/IHx+QMFD5ZM</latexit><latexit sha1_base64="JZCaNuWBzvB4Fhg6CQVX04IWYps=">AAACBXicbVC7SgNBFL0bXzG+Vi21GAyCVdwNgjZCICksLKKYB2TXZXYySYbMPpiZFcKSxsZfsbFQxNZ/sPNvnCRbaOKBC4dz7uXee/yYM6ks69vILS2vrK7l1wsbm1vbO+buXlNGiSC0QSIeibaPJeUspA3FFKftWFAc+Jy2/GF14rceqJAsCu/UKKZugPsh6zGClZY88/AaXaKmlzoEc3RbHaNT5NQoV/i+hoaeWbRK1hRokdgZKUKGumd+Od2IJAENFeFYyo5txcpNsVCMcDouOImkMSZD3KcdTUMcUOmm0y/G6FgrXdSLhK5Qoan6eyLFgZSjwNedAVYDOe9NxP+8TqJ6F27KwjhRNCSzRb2EIxWhSSSoywQlio80wUQwfSsiAywwUTq4gg7Bnn95kTTLJdsq2TdnxUo5iyMPB3AEJ2DDOVTgCurQAAKP8Ayv8GY8GS/Gu/Exa80Z2cw+/IHx+QMFD5ZM</latexit>










reciprocal space


�2k<latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit><latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit><latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit><latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit>

A↵ = L↵a↵<latexit sha1_base64="M5lGwLQ5BKmAnGck9fVuMm1+Zf8=">AAACIXicbVDLSsNAFJ3UV62vqks3g0VwVZIi2I1QcePCRQX7gCaEm8mkHTp5MDMRSuivuPFX3LhQpDvxZ5z0Abb1wjCHc87l3nu8hDOpTPPbKGxsbm3vFHdLe/sHh0fl45O2jFNBaIvEPBZdDyTlLKItxRSn3URQCD1OO97wLtc7z1RIFkdPapRQJ4R+xAJGQGnKLdcz24u5L0eh/vDt2LWBJwPAN/hhAZccsHC45YpZNaeF14E1BxU0r6Zbnth+TNKQRopwkLJnmYlyMhCKEU7HJTuVNAEyhD7taRhBSKWTTS8c4wvN+DiIhX6RwlP2b0cGocw31M4Q1ECuajn5n9ZLVVB3MhYlqaIRmQ0KUo5VjPO4sM8EJYqPNAAimN4VkwEIIEqHWtIhWKsnr4N2rWqZVevxqtKozeMoojN0ji6Rha5RA92jJmohgl7QG/pAn8ar8W58GZOZtWDMe07RUhk/v/JSo+8=</latexit><latexit sha1_base64="M5lGwLQ5BKmAnGck9fVuMm1+Zf8=">AAACIXicbVDLSsNAFJ3UV62vqks3g0VwVZIi2I1QcePCRQX7gCaEm8mkHTp5MDMRSuivuPFX3LhQpDvxZ5z0Abb1wjCHc87l3nu8hDOpTPPbKGxsbm3vFHdLe/sHh0fl45O2jFNBaIvEPBZdDyTlLKItxRSn3URQCD1OO97wLtc7z1RIFkdPapRQJ4R+xAJGQGnKLdcz24u5L0eh/vDt2LWBJwPAN/hhAZccsHC45YpZNaeF14E1BxU0r6Zbnth+TNKQRopwkLJnmYlyMhCKEU7HJTuVNAEyhD7taRhBSKWTTS8c4wvN+DiIhX6RwlP2b0cGocw31M4Q1ECuajn5n9ZLVVB3MhYlqaIRmQ0KUo5VjPO4sM8EJYqPNAAimN4VkwEIIEqHWtIhWKsnr4N2rWqZVevxqtKozeMoojN0ji6Rha5RA92jJmohgl7QG/pAn8ar8W58GZOZtWDMe07RUhk/v/JSo+8=</latexit><latexit sha1_base64="M5lGwLQ5BKmAnGck9fVuMm1+Zf8=">AAACIXicbVDLSsNAFJ3UV62vqks3g0VwVZIi2I1QcePCRQX7gCaEm8mkHTp5MDMRSuivuPFX3LhQpDvxZ5z0Abb1wjCHc87l3nu8hDOpTPPbKGxsbm3vFHdLe/sHh0fl45O2jFNBaIvEPBZdDyTlLKItxRSn3URQCD1OO97wLtc7z1RIFkdPapRQJ4R+xAJGQGnKLdcz24u5L0eh/vDt2LWBJwPAN/hhAZccsHC45YpZNaeF14E1BxU0r6Zbnth+TNKQRopwkLJnmYlyMhCKEU7HJTuVNAEyhD7taRhBSKWTTS8c4wvN+DiIhX6RwlP2b0cGocw31M4Q1ECuajn5n9ZLVVB3MhYlqaIRmQ0KUo5VjPO4sM8EJYqPNAAimN4VkwEIIEqHWtIhWKsnr4N2rWqZVevxqtKozeMoojN0ji6Rha5RA92jJmohgl7QG/pAn8ar8W58GZOZtWDMe07RUhk/v/JSo+8=</latexit><latexit sha1_base64="M5lGwLQ5BKmAnGck9fVuMm1+Zf8=">AAACIXicbVDLSsNAFJ3UV62vqks3g0VwVZIi2I1QcePCRQX7gCaEm8mkHTp5MDMRSuivuPFX3LhQpDvxZ5z0Abb1wjCHc87l3nu8hDOpTPPbKGxsbm3vFHdLe/sHh0fl45O2jFNBaIvEPBZdDyTlLKItxRSn3URQCD1OO97wLtc7z1RIFkdPapRQJ4R+xAJGQGnKLdcz24u5L0eh/vDt2LWBJwPAN/hhAZccsHC45YpZNaeF14E1BxU0r6Zbnth+TNKQRopwkLJnmYlyMhCKEU7HJTuVNAEyhD7taRhBSKWTTS8c4wvN+DiIhX6RwlP2b0cGocw31M4Q1ECuajn5n9ZLVVB3MhYlqaIRmQ0KUo5VjPO4sM8EJYqPNAAimN4VkwEIIEqHWtIhWKsnr4N2rWqZVevxqtKozeMoojN0ji6Rha5RA92jJmohgl7QG/pAn8ar8W58GZOZtWDMe07RUhk/v/JSo+8=</latexit>

f(x) =X

k

eikxfk

<latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="feYzCSbqj61LrrcowietRi1DSP8=">AAAB5HicbVDLSgNBEOz1GddX9OplMAiewq4XPQpePEYwD0hCmJ3tTYbMzi4zvUJY8gMevIh49Zu8+TdOHqAmFjQUVd10d0W5kpaC4Mvb2Nza3tmt7Pn7B/7h0XH1pGWzwghsikxlphNxi0pqbJIkhZ3cIE8jhe1ofDfz209orMz0I01y7Kd8qGUiBScnNQbVWlAP5mDrJFySGiwxqH724kwUKWoSilvbDYOc+iU3JIXCqd8rLOZcjPkQu45qnqLtl/Mzp+zCKTFLMuNKE5urvydKnlo7SSPXmXIa2VVvJv7ndQtKbvql1HlBqMViUVIoRhmb/cxiaVCQmjjChZHuViZG3HBBLhnfZRCufrxOWlf1MKiHDz9hQAXO4BwuIYRruIV7aEATBMTwDK/eyHvx3rz3ReOGt5w4hT/wPr4BGDqLAg==</latexit><latexit sha1_base64="AuLdStPmI8ab4Bs58lr8uVqp4kY=">AAACSHicbVBLS8NAGPwSX7W+qh69LBahXkriRS+C4MVjBfuAJobNZtMu3TzY3Ygl5Od58ejN3+DFgyLe3D5Amzqw7DDzDfvt+ClnUlnWq2GurK6tb1Q2q1vbO7t7tf2DjkwyQWibJDwRPR9LyllM24opTnupoDjyOe36o+uJ332gQrIkvlPjlLoRHsQsZAQrLXk1L2zkjp/wQI4jfaHH4hRdOjKLvAV5VCB6nzNUEkvRAoXlmFerW01rCrRM7Dmpwxwtr/biBAnJIhorwrGUfdtKlZtjoRjhtKg6maQpJiM8oH1NYxxR6ebTIgp0opUAhYnQJ1Zoqv5N5DiSk9X0ZITVUJa9ifif189UeOHmLE4zRWMyeyjMOFIJmrSKAiYoUXysCSaC6V0RGWKBidLdV3UJdvnLy6Rz1rStpn372wZU4AiOoQE2nMMV3EAL2kDgCd7gAz6NZ+Pd+DK+Z6OmMc8cwgJM8wfxa7PR</latexit><latexit sha1_base64="AuLdStPmI8ab4Bs58lr8uVqp4kY=">AAACSHicbVBLS8NAGPwSX7W+qh69LBahXkriRS+C4MVjBfuAJobNZtMu3TzY3Ygl5Od58ejN3+DFgyLe3D5Amzqw7DDzDfvt+ClnUlnWq2GurK6tb1Q2q1vbO7t7tf2DjkwyQWibJDwRPR9LyllM24opTnupoDjyOe36o+uJ332gQrIkvlPjlLoRHsQsZAQrLXk1L2zkjp/wQI4jfaHH4hRdOjKLvAV5VCB6nzNUEkvRAoXlmFerW01rCrRM7Dmpwxwtr/biBAnJIhorwrGUfdtKlZtjoRjhtKg6maQpJiM8oH1NYxxR6ebTIgp0opUAhYnQJ1Zoqv5N5DiSk9X0ZITVUJa9ifif189UeOHmLE4zRWMyeyjMOFIJmrSKAiYoUXysCSaC6V0RGWKBidLdV3UJdvnLy6Rz1rStpn372wZU4AiOoQE2nMMV3EAL2kDgCd7gAz6NZ+Pd+DK+Z6OmMc8cwgJM8wfxa7PR</latexit><latexit sha1_base64="EAm3E+/NYU38XTpUS829GLi+DFk=">AAACSHicbVA7T8MwGHTCq5RXgJHFokIqS5WwwIJUiYWxSPQhNSVyHKe1aieR7SCqKD+PhZGN38DCAEJsOG0GmnKS5dPdd/Ln8xNGpbLtN8NcW9/Y3Kpt13d29/YPrMOjnoxTgUkXxywWAx9JwmhEuooqRgaJIIj7jPT96U3h9x+JkDSO7tUsISOOxhENKUZKS57lhc3M9WMWyBnXF3zKz+G1K1PuLcnTHJKHjMKKWInmMKzGPKtht+w54CpxStIAJTqe9eoGMU45iRRmSMqhYydqlCGhKGYkr7upJAnCUzQmQ00jxIkcZfMicnimlQCGsdAnUnCu/k1kiMtiNT3JkZrIqleI/3nDVIVXo4xGSapIhBcPhSmDKoZFqzCggmDFZpogLKjeFeIJEggr3X1dl+BUv7xKehctx245d3ajfVHWUQMn4BQ0gQMuQRvcgg7oAgyewTv4BF/Gi/FhfBs/i1HTKDPHYAmm+QsUBLRD</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit><latexit sha1_base64="yLiuDsUMBFPmWrI+dI76H8kNhuw=">AAACSHicbVDNS8MwHE3r15xfVY9egkOYl9EOQS/CwIvHCe4D1lrSNN3C0g+SVBylf54Xj978G7x4UMSb6daD63wQ8njv98gvz0sYFdI03zR9bX1jc6u2Xd/Z3ds/MA6P+iJOOSY9HLOYDz0kCKMR6UkqGRkmnKDQY2TgTW8Kf/BIuKBxdC9nCXFCNI5oQDGSSnINN2hmthczX8xCdcGn/Bxe2yIN3SV5mkPykFFYESvRHAbVmGs0zJY5B1wlVkkaoETXNV5tP8ZpSCKJGRJiZJmJdDLEJcWM5HU7FSRBeIrGZKRohEIinGxeRA7PlOLDIObqRBLO1b+JDIWiWE1NhkhORNUrxP+8USqDKyejUZJKEuHFQ0HKoIxh0Sr0KSdYspkiCHOqdoV4gjjCUnVfVyVY1S+vkn67ZZkt6+6i0WmXddTACTgFTWCBS9ABt6ALegCDZ/AOPsGX9qJ9aN/az2JU18rMMViCrv8CFUS0Rw==</latexit>

superlattice and reciprocal superlattice


ea1<latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit><latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit><latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit><latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit> eb1

<latexit sha1_base64="DzI5PRLCSO7Qlf9GnARwFxzTdxc=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIf7lfFJ7tmQ2rac2Al4ldkQaq0PHMLydIaBZBrCgnUg5tK1VuToRilENRdzIJKaETMoKhpjGJQLr57IoCn2klwGEi9IsVnqm/O3ISyXI7XRkRNZaLXin+5w0zFbbcnMVppiCm80FhxrFKcBkJDpgAqvhUE0IF07tiOiaCUKWDq+sQ7MWTl0nvomlbTfv2stFuVXHU0DE6RefIRleojW5QB3URRY/oGb2iN+PJeDHejY956YpR9RyhPzA+fwCrs5lF</latexit><latexit sha1_base64="DzI5PRLCSO7Qlf9GnARwFxzTdxc=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIf7lfFJ7tmQ2rac2Al4ldkQaq0PHMLydIaBZBrCgnUg5tK1VuToRilENRdzIJKaETMoKhpjGJQLr57IoCn2klwGEi9IsVnqm/O3ISyXI7XRkRNZaLXin+5w0zFbbcnMVppiCm80FhxrFKcBkJDpgAqvhUE0IF07tiOiaCUKWDq+sQ7MWTl0nvomlbTfv2stFuVXHU0DE6RefIRleojW5QB3URRY/oGb2iN+PJeDHejY956YpR9RyhPzA+fwCrs5lF</latexit><latexit sha1_base64="DzI5PRLCSO7Qlf9GnARwFxzTdxc=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIf7lfFJ7tmQ2rac2Al4ldkQaq0PHMLydIaBZBrCgnUg5tK1VuToRilENRdzIJKaETMoKhpjGJQLr57IoCn2klwGEi9IsVnqm/O3ISyXI7XRkRNZaLXin+5w0zFbbcnMVppiCm80FhxrFKcBkJDpgAqvhUE0IF07tiOiaCUKWDq+sQ7MWTl0nvomlbTfv2stFuVXHU0DE6RefIRleojW5QB3URRY/oGb2iN+PJeDHejY956YpR9RyhPzA+fwCrs5lF</latexit><latexit sha1_base64="DzI5PRLCSO7Qlf9GnARwFxzTdxc=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIf7lfFJ7tmQ2rac2Al4ldkQaq0PHMLydIaBZBrCgnUg5tK1VuToRilENRdzIJKaETMoKhpjGJQLr57IoCn2klwGEi9IsVnqm/O3ISyXI7XRkRNZaLXin+5w0zFbbcnMVppiCm80FhxrFKcBkJDpgAqvhUE0IF07tiOiaCUKWDq+sQ7MWTl0nvomlbTfv2stFuVXHU0DE6RefIRleojW5QB3URRY/oGb2iN+PJeDHejY956YpR9RyhPzA+fwCrs5lF</latexit>

eb2<latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit><latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit><latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit><latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit>

ea2<latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit><latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit><latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit><latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit>






eR<latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit>

eG<latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit><latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit><latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit><latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit>

R = eR+ r<latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit>


SC<latexit sha1_base64="n76Yn0otP2pAO9vKIdaXIhlVSnc=">AAAB7nicdVDLSgMxFL1TX7W+qi7dBIvgasjUqa27QjcuK9oHtEPJpGkbmnmQZIQy9CPcuFDErd/jzr8x01ZQ0QOBk3Pu5d57/FhwpTH+sHJr6xubW/ntws7u3v5B8fCoraJEUtaikYhk1yeKCR6yluZasG4sGQl8wTr+tJH5nXsmFY/COz2LmReQcchHnBJtpE6fEoFuG4NiCdu4ii+cMsK2W3Grlxkxf/eqghwbL1CCFZqD4nt/GNEkYKGmgijVc3CsvZRIzalg80I/USwmdErGrGdoSAKmvHSx7hydGWWIRpE0L9RooX7vSEmg1CzwTWVA9ET99jLxL6+X6FHNS3kYJ5qFdDlolAikI5TdjoZcMqrFzBBCJTe7IjohklBtEiqYEL4uRf+Tdtl2sO3cuKV6bRVHHk7gFM7BgSrU4Rqa0AIKU3iAJ3i2YuvRerFel6U5a9VzDD9gvX0C2U2PNw==</latexit><latexit sha1_base64="n76Yn0otP2pAO9vKIdaXIhlVSnc=">AAAB7nicdVDLSgMxFL1TX7W+qi7dBIvgasjUqa27QjcuK9oHtEPJpGkbmnmQZIQy9CPcuFDErd/jzr8x01ZQ0QOBk3Pu5d57/FhwpTH+sHJr6xubW/ntws7u3v5B8fCoraJEUtaikYhk1yeKCR6yluZasG4sGQl8wTr+tJH5nXsmFY/COz2LmReQcchHnBJtpE6fEoFuG4NiCdu4ii+cMsK2W3Grlxkxf/eqghwbL1CCFZqD4nt/GNEkYKGmgijVc3CsvZRIzalg80I/USwmdErGrGdoSAKmvHSx7hydGWWIRpE0L9RooX7vSEmg1CzwTWVA9ET99jLxL6+X6FHNS3kYJ5qFdDlolAikI5TdjoZcMqrFzBBCJTe7IjohklBtEiqYEL4uRf+Tdtl2sO3cuKV6bRVHHk7gFM7BgSrU4Rqa0AIKU3iAJ3i2YuvRerFel6U5a9VzDD9gvX0C2U2PNw==</latexit><latexit sha1_base64="n76Yn0otP2pAO9vKIdaXIhlVSnc=">AAAB7nicdVDLSgMxFL1TX7W+qi7dBIvgasjUqa27QjcuK9oHtEPJpGkbmnmQZIQy9CPcuFDErd/jzr8x01ZQ0QOBk3Pu5d57/FhwpTH+sHJr6xubW/ntws7u3v5B8fCoraJEUtaikYhk1yeKCR6yluZasG4sGQl8wTr+tJH5nXsmFY/COz2LmReQcchHnBJtpE6fEoFuG4NiCdu4ii+cMsK2W3Grlxkxf/eqghwbL1CCFZqD4nt/GNEkYKGmgijVc3CsvZRIzalg80I/USwmdErGrGdoSAKmvHSx7hydGWWIRpE0L9RooX7vSEmg1CzwTWVA9ET99jLxL6+X6FHNS3kYJ5qFdDlolAikI5TdjoZcMqrFzBBCJTe7IjohklBtEiqYEL4uRf+Tdtl2sO3cuKV6bRVHHk7gFM7BgSrU4Rqa0AIKU3iAJ3i2YuvRerFel6U5a9VzDD9gvX0C2U2PNw==</latexit><latexit sha1_base64="n76Yn0otP2pAO9vKIdaXIhlVSnc=">AAAB7nicdVDLSgMxFL1TX7W+qi7dBIvgasjUqa27QjcuK9oHtEPJpGkbmnmQZIQy9CPcuFDErd/jzr8x01ZQ0QOBk3Pu5d57/FhwpTH+sHJr6xubW/ntws7u3v5B8fCoraJEUtaikYhk1yeKCR6yluZasG4sGQl8wTr+tJH5nXsmFY/COz2LmReQcchHnBJtpE6fEoFuG4NiCdu4ii+cMsK2W3Grlxkxf/eqghwbL1CCFZqD4nt/GNEkYKGmgijVc3CsvZRIzalg80I/USwmdErGrGdoSAKmvHSx7hydGWWIRpE0L9RooX7vSEmg1CzwTWVA9ET99jLxL6+X6FHNS3kYJ5qFdDlolAikI5TdjoZcMqrFzBBCJTe7IjohklBtEiqYEL4uRf+Tdtl2sO3cuKV6bRVHHk7gFM7BgSrU4Rqa0AIKU3iAJ3i2YuvRerFel6U5a9VzDD9gvX0C2U2PNw==</latexit>



RSC<latexit sha1_base64="+uII9F5ToJyAR60F2xalf8d70q8=">AAAB73icdVDLSgMxFM3UV62vqks3wSK4GjJ1auuu0I3L+ugD2qFk0kwbmsmMSUYoQ3/CjQtF3Po77vwbM20FFT0QODnnXu69x485UxqhDyu3srq2vpHfLGxt7+zuFfcP2ipKJKEtEvFIdn2sKGeCtjTTnHZjSXHoc9rxJ43M79xTqVgkbvU0pl6IR4IFjGBtpG6fYA6vbxqDYgnZqIrOnDJEtltxq+cZMX/3ogIdG81RAks0B8X3/jAiSUiFJhwr1XNQrL0US80Ip7NCP1E0xmSCR7RnqMAhVV4633cGT4wyhEEkzRMaztXvHSkOlZqGvqkMsR6r314m/uX1Eh3UvJSJONFUkMWgIOFQRzA7Hg6ZpETzqSGYSGZ2hWSMJSbaRFQwIXxdCv8n7bLtINu5ckv12jKOPDgCx+AUOKAK6uASNEELEMDBA3gCz9ad9Wi9WK+L0py17DkEP2C9fQJ85I+T</latexit><latexit sha1_base64="+uII9F5ToJyAR60F2xalf8d70q8=">AAAB73icdVDLSgMxFM3UV62vqks3wSK4GjJ1auuu0I3L+ugD2qFk0kwbmsmMSUYoQ3/CjQtF3Po77vwbM20FFT0QODnnXu69x485UxqhDyu3srq2vpHfLGxt7+zuFfcP2ipKJKEtEvFIdn2sKGeCtjTTnHZjSXHoc9rxJ43M79xTqVgkbvU0pl6IR4IFjGBtpG6fYA6vbxqDYgnZqIrOnDJEtltxq+cZMX/3ogIdG81RAks0B8X3/jAiSUiFJhwr1XNQrL0US80Ip7NCP1E0xmSCR7RnqMAhVV4633cGT4wyhEEkzRMaztXvHSkOlZqGvqkMsR6r314m/uX1Eh3UvJSJONFUkMWgIOFQRzA7Hg6ZpETzqSGYSGZ2hWSMJSbaRFQwIXxdCv8n7bLtINu5ckv12jKOPDgCx+AUOKAK6uASNEELEMDBA3gCz9ad9Wi9WK+L0py17DkEP2C9fQJ85I+T</latexit><latexit sha1_base64="+uII9F5ToJyAR60F2xalf8d70q8=">AAAB73icdVDLSgMxFM3UV62vqks3wSK4GjJ1auuu0I3L+ugD2qFk0kwbmsmMSUYoQ3/CjQtF3Po77vwbM20FFT0QODnnXu69x485UxqhDyu3srq2vpHfLGxt7+zuFfcP2ipKJKEtEvFIdn2sKGeCtjTTnHZjSXHoc9rxJ43M79xTqVgkbvU0pl6IR4IFjGBtpG6fYA6vbxqDYgnZqIrOnDJEtltxq+cZMX/3ogIdG81RAks0B8X3/jAiSUiFJhwr1XNQrL0US80Ip7NCP1E0xmSCR7RnqMAhVV4633cGT4wyhEEkzRMaztXvHSkOlZqGvqkMsR6r314m/uX1Eh3UvJSJONFUkMWgIOFQRzA7Hg6ZpETzqSGYSGZ2hWSMJSbaRFQwIXxdCv8n7bLtINu5ckv12jKOPDgCx+AUOKAK6uASNEELEMDBA3gCz9ad9Wi9WK+L0py17DkEP2C9fQJ85I+T</latexit><latexit sha1_base64="+uII9F5ToJyAR60F2xalf8d70q8=">AAAB73icdVDLSgMxFM3UV62vqks3wSK4GjJ1auuu0I3L+ugD2qFk0kwbmsmMSUYoQ3/CjQtF3Po77vwbM20FFT0QODnnXu69x485UxqhDyu3srq2vpHfLGxt7+zuFfcP2ipKJKEtEvFIdn2sKGeCtjTTnHZjSXHoc9rxJ43M79xTqVgkbvU0pl6IR4IFjGBtpG6fYA6vbxqDYgnZqIrOnDJEtltxq+cZMX/3ogIdG81RAks0B8X3/jAiSUiFJhwr1XNQrL0US80Ip7NCP1E0xmSCR7RnqMAhVV4633cGT4wyhEEkzRMaztXvHSkOlZqGvqkMsR6r314m/uX1Eh3UvJSJONFUkMWgIOFQRzA7Hg6ZpETzqSGYSGZ2hWSMJSbaRFQwIXxdCv8n7bLtINu5ckv12jKOPDgCx+AUOKAK6uASNEELEMDBA3gCz9ad9Wi9WK+L0py17DkEP2C9fQJ85I+T</latexit>


reciprocal space


r<latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit>

ek<latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit><latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit><latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit><latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit>

k = eG+ ek<latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit><latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit><latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit><latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit>

R =DX

↵=1

i↵a↵

<latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit><latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit><latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit><latexit sha1_base64="xhJyUCK36C2l8EffT3FH6yDlyMU=">AAACLHicbVDLSgMxFM3UV62vqks3wSK4KjNF0E2hUBcuq9gHdMbhTiZtQzMPkoxQhvkgN/6KIC4s4tbvMG1nUVsPhJycey6593gxZ1KZ5tQobGxube8Ud0t7+weHR+Xjk46MEkFom0Q8Ej0PJOUspG3FFKe9WFAIPE673rg5q3efqZAsCh/VJKZOAMOQDRgBpSW33ExtL+K+nAT6wg8ZrtsyCdzUBh6PoG5lT+lthpm7eOMlcwpZrrrlilk158DrxMpJBeVoueV3249IEtBQEQ5S9i0zVk4KQjHCaVayE0ljIGMY0r6mIQRUOul82QxfaMXHg0joEyo8V5c7UgjkbELtDECN5GptJv5X6ydqcOOkLIwTRUOy+GiQcKwiPEsO+0xQovhEEyCC6VkxGYEAonS+JR2CtbryOunUqpZZte6vKo1aHkcRnaFzdIksdI0a6A61UBsR9ILe0CeaGq/Gh/FlfC+sBSPvOUV/YPz8AsqCqRw=</latexit>

G =DX

�=1

j�b�<latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="feYzCSbqj61LrrcowietRi1DSP8=">AAAB5HicbVDLSgNBEOz1GddX9OplMAiewq4XPQpePEYwD0hCmJ3tTYbMzi4zvUJY8gMevIh49Zu8+TdOHqAmFjQUVd10d0W5kpaC4Mvb2Nza3tmt7Pn7B/7h0XH1pGWzwghsikxlphNxi0pqbJIkhZ3cIE8jhe1ofDfz209orMz0I01y7Kd8qGUiBScnNQbVWlAP5mDrJFySGiwxqH724kwUKWoSilvbDYOc+iU3JIXCqd8rLOZcjPkQu45qnqLtl/Mzp+zCKTFLMuNKE5urvydKnlo7SSPXmXIa2VVvJv7ndQtKbvql1HlBqMViUVIoRhmb/cxiaVCQmjjChZHuViZG3HBBLhnfZRCufrxOWlf1MKiHDz9hQAXO4BwuIYRruIV7aEATBMTwDK/eyHvx3rz3ReOGt5w4hT/wPr4BGDqLAg==</latexit><latexit sha1_base64="iC3FV0sIqBK97Y3uXBKAMCLRIj0=">AAACKXicbVDLSgMxFL3js9ZX1aWbYBFclRk3uikUFHRZwT6gHYdMmmljM5MhyQhlmN9x46+4UVDUrT9ipi1YWw+EnJx7Lrn3+DFnStv2p7W0vLK6tl7YKG5ube/slvb2m0okktAGEVzIto8V5SyiDc00p+1YUhz6nLb84UVebz1QqZiIbvUopm6I+xELGMHaSF6plnZ9wXtqFJoLXWWo2lVJ6BmValx1srv0MkP33viJZqypn01Er1S2K/YYaJE4U1KGKepe6bXbEyQJaaQJx0p1HDvWboqlZoTTrNhNFI0xGeI+7Rga4ZAqNx1vmqFjo/RQIKQ5kUZjdbYjxaHKBzTOEOuBmq/l4n+1TqKDczdlUZxoGpHJR0HCkRYojw31mKRE85EhmEhmZkVkgCUm2oRbNCE48ysvkuZpxbErzs1vGlCAQziCE3DgDGpwDXVoAIFHeIY3eLeerBfrw/qaWJesac8B/IH1/QP3lqdB</latexit><latexit sha1_base64="iC3FV0sIqBK97Y3uXBKAMCLRIj0=">AAACKXicbVDLSgMxFL3js9ZX1aWbYBFclRk3uikUFHRZwT6gHYdMmmljM5MhyQhlmN9x46+4UVDUrT9ipi1YWw+EnJx7Lrn3+DFnStv2p7W0vLK6tl7YKG5ube/slvb2m0okktAGEVzIto8V5SyiDc00p+1YUhz6nLb84UVebz1QqZiIbvUopm6I+xELGMHaSF6plnZ9wXtqFJoLXWWo2lVJ6BmValx1srv0MkP33viJZqypn01Er1S2K/YYaJE4U1KGKepe6bXbEyQJaaQJx0p1HDvWboqlZoTTrNhNFI0xGeI+7Rga4ZAqNx1vmqFjo/RQIKQ5kUZjdbYjxaHKBzTOEOuBmq/l4n+1TqKDczdlUZxoGpHJR0HCkRYojw31mKRE85EhmEhmZkVkgCUm2oRbNCE48ysvkuZpxbErzs1vGlCAQziCE3DgDGpwDXVoAIFHeIY3eLeerBfrw/qaWJesac8B/IH1/QP3lqdB</latexit><latexit sha1_base64="Fh7k5r2yFq73WxqVPul8HWDl+90=">AAACKXicbVBNS8MwGE79nPOr6tFLcAieRruLXgYDBT1OcB+wzpKm6RaXNiVJhVH6d7z4V7woKOrVP2La9TA3Hwh58rzPS9738WJGpbKsL2NldW19Y7OyVd3e2d3bNw8Ou5InApMO5oyLvockYTQiHUUVI/1YEBR6jPS8yWVe7z0SISmP7tQ0JsMQjSIaUIyUllyzlToeZ76chvqC1xlsOjIJXa0ShZp2dp9eZfDBLZ5wzpp62Ux0zZpVtwrAZWKXpAZKtF3zzfE5TkISKcyQlAPbitUwRUJRzEhWdRJJYoQnaEQGmkYoJHKYFptm8FQrPgy40CdSsFDnO1IUynxA7QyRGsvFWi7+VxskKrgYpjSKE0UiPPsoSBhUHOaxQZ8KghWbaoKwoHpWiMdIIKx0uFUdgr248jLpNuq2VbdvrVqrUcZRAcfgBJwBG5yDFrgBbdABGDyBF/AOPoxn49X4NL5n1hWj7DkCf2D8/AIaL6ez</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit><latexit sha1_base64="otcFyNmTvyWbmcipQJ6sL83smaw=">AAACKXicbVDLSsNAFJ3UV62vqEs3g0VwVZIi6KZQUNBlBfuAJobJZNKOnTyYmQgl5Hfc+CtuFBR16484SbOorQeGOXPuucy9x40ZFdIwvrTKyura+kZ1s7a1vbO7p+8f9ESUcEy6OGIRH7hIEEZD0pVUMjKIOUGBy0jfnVzm9f4j4YJG4Z2cxsQO0CikPsVIKsnR26nlRswT00Bd8DqDLUskgaNUIlHLzO7Tqww+OMUTzllTN5uJjl43GkYBuEzMktRBiY6jv1lehJOAhBIzJMTQNGJpp4hLihnJalYiSIzwBI3IUNEQBUTYabFpBk+U4kE/4uqEEhbqfEeKApEPqJwBkmOxWMvF/2rDRPoXdkrDOJEkxLOP/IRBGcE8NuhRTrBkU0UQ5lTNCvEYcYSlCremQjAXV14mvWbDNBrm7Vm93SzjqIIjcAxOgQnOQRvcgA7oAgyewAt4Bx/as/aqfWrfM2tFK3sOwR9oP78bb6e3</latexit>

exp(iGR) = 1

<latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit><latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit><latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit><latexit sha1_base64="ma4KXyZdxcCKLP84ygp046GZP2c=">AAACFXicbVBNS8MwGE79nPOr6tFLcAgTZLRD0Isw8KDHKe4D1jLSNN3C0rQkqThK/4QX/4oXD4p4Fbz5b0y3HtzmAyFPnud9yfs+XsyoVJb1Yywtr6yurZc2yptb2zu75t5+W0aJwKSFIxaJrockYZSTlqKKkW4sCAo9Rjre6Cr3Ow9ESBrxezWOiRuiAacBxUhpqW+eOuQxhlUKU8eLmC/Hob7gdTb7vstO4CW0+2bFqlkTwEViF6QCCjT75rfjRzgJCVeYISl7thUrN0VCUcxIVnYSSWKER2hAeppyFBLpppOtMnisFR8GkdCHKzhR/3akKJT5fLoyRGoo571c/M/rJSq4cFPK40QRjqcfBQmDKoJ5RNCngmDFxpogLKieFeIhEggrHWRZh2DPr7xI2vWabdXs27NKo17EUQKH4AhUgQ3OQQPcgCZoAQyewAt4A+/Gs/FqfBif09Ilo+g5ADMwvn4BiGWdsw==</latexit>

a↵b� = 2⇡�↵�<latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit><latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit><latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit><latexit sha1_base64="PIuCBlbS5rBFQi5usbftLYTaHuM=">AAACMXicbVDLSgNBEJz1GeMr6tHLYBA8hd0g6EUIePEYwRghG5be2U4yOPtgplcIS37Ji38iXjwo4tWfcBL3YKINwxRV1XR3hZmShlz31VlaXlldW69sVDe3tnd2a3v7tybNtcCOSFWq70IwqGSCHZKk8C7TCHGosBveX0717gNqI9PkhsYZ9mMYJnIgBZClgtpV4Yepisw4th+HSeCDykbA5+jQ0iES8Ave9DPpR6gIgqK0zqRJUKu7DXdW/C/wSlBnZbWD2rMfpSKPMSGhwJie52bUL0CTFAonVT83mIG4hyH2LEwgRtMvZhdP+LFlIj5ItX0J8Rn7u6OA2EyXt84YaGQWtSn5n9bLaXDeL2SS5YSJ+Bk0yBWnlE/j45HUKEiNLQChpd2VixFoEGRDrtoQvMWT/4LbZsNzG971ab3VLOOosEN2xE6Yx85Yi12xNuswwR7ZC3tj786T8+p8OJ8/1iWn7Dlgc+V8fQNmfqrX</latexit>

e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit>e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> e<latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit><latexit sha1_base64="L2HMg7S4elMAsLFbjHmi0cEI+co=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWw2k3bpZhN2J0oJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEph0HW/ndLG5tb2Tnm3srd/cHhUPT7pmCTTHNo8kYnuBcyAFAraKFBCL9XA4kBCN5jczv3uI2gjEvWA0xT8mI2UiARnaKXB4EmEgEKGkM+G1Zpbdxeg68QrSI0UaA2rX4Mw4VkMCrlkxvQ9N0U/ZxoFlzCrDDIDKeMTNoK+pYrFYPx8cfOMXlglpFGibSmkC/X3RM5iY6ZxYDtjhmOz6s3F/7x+htGNnwuVZgiKLxdFmaSY0HkANBQaOMqpJYxrYW+lfMw042hjqtgQvNWX10mnUffcund/VWs2ijjK5Iyck0vikWvSJHekRdqEk5Q8k1fy5mTOi/PufCxbS04xc0r+wPn8AYgakfE=</latexit> VSCVRSC = (2⇡)D<latexit sha1_base64="Ir/RQZvO0o9kVEjt6fmdbD2Nryw=">AAACDHicbVDLSgMxFL1TX7W+qi7dBItQN2WmCLoRCnXhsj76gM5YMmmmDc08SDJCGfoBbvwVNy4UcesHuPNvzLQjaOuBwMk555Lc40acSWWaX0ZuaXlldS2/XtjY3NreKe7utWQYC0KbJOSh6LhYUs4C2lRMcdqJBMW+y2nbHdVTv31PhWRhcKvGEXV8PAiYxwhWWuoVS61eYhPM0U19gn74dXo5R+WqHbHjuwudMivmFGiRWBkpQYZGr/hp90MS+zRQhGMpu5YZKSfBQjHC6aRgx5JGmIzwgHY1DbBPpZNMl5mgI630kRcKfQKFpurviQT7Uo59Vyd9rIZy3kvF/7xurLwzJ2FBFCsakNlDXsyRClHaDOozQYniY00wEUz/FZEhFpgo3V9Bl2DNr7xIWtWKZVasq5NSrZrVkYcDOIQyWHAKNbiEBjSBwAM8wQu8Go/Gs/FmvM+iOSOb2Yc/MD6+AUFMmSA=</latexit><latexit sha1_base64="Ir/RQZvO0o9kVEjt6fmdbD2Nryw=">AAACDHicbVDLSgMxFL1TX7W+qi7dBItQN2WmCLoRCnXhsj76gM5YMmmmDc08SDJCGfoBbvwVNy4UcesHuPNvzLQjaOuBwMk555Lc40acSWWaX0ZuaXlldS2/XtjY3NreKe7utWQYC0KbJOSh6LhYUs4C2lRMcdqJBMW+y2nbHdVTv31PhWRhcKvGEXV8PAiYxwhWWuoVS61eYhPM0U19gn74dXo5R+WqHbHjuwudMivmFGiRWBkpQYZGr/hp90MS+zRQhGMpu5YZKSfBQjHC6aRgx5JGmIzwgHY1DbBPpZNMl5mgI630kRcKfQKFpurviQT7Uo59Vyd9rIZy3kvF/7xurLwzJ2FBFCsakNlDXsyRClHaDOozQYniY00wEUz/FZEhFpgo3V9Bl2DNr7xIWtWKZVasq5NSrZrVkYcDOIQyWHAKNbiEBjSBwAM8wQu8Go/Gs/FmvM+iOSOb2Yc/MD6+AUFMmSA=</latexit><latexit sha1_base64="Ir/RQZvO0o9kVEjt6fmdbD2Nryw=">AAACDHicbVDLSgMxFL1TX7W+qi7dBItQN2WmCLoRCnXhsj76gM5YMmmmDc08SDJCGfoBbvwVNy4UcesHuPNvzLQjaOuBwMk555Lc40acSWWaX0ZuaXlldS2/XtjY3NreKe7utWQYC0KbJOSh6LhYUs4C2lRMcdqJBMW+y2nbHdVTv31PhWRhcKvGEXV8PAiYxwhWWuoVS61eYhPM0U19gn74dXo5R+WqHbHjuwudMivmFGiRWBkpQYZGr/hp90MS+zRQhGMpu5YZKSfBQjHC6aRgx5JGmIzwgHY1DbBPpZNMl5mgI630kRcKfQKFpurviQT7Uo59Vyd9rIZy3kvF/7xurLwzJ2FBFCsakNlDXsyRClHaDOozQYniY00wEUz/FZEhFpgo3V9Bl2DNr7xIWtWKZVasq5NSrZrVkYcDOIQyWHAKNbiEBjSBwAM8wQu8Go/Gs/FmvM+iOSOb2Yc/MD6+AUFMmSA=</latexit><latexit sha1_base64="Ir/RQZvO0o9kVEjt6fmdbD2Nryw=">AAACDHicbVDLSgMxFL1TX7W+qi7dBItQN2WmCLoRCnXhsj76gM5YMmmmDc08SDJCGfoBbvwVNy4UcesHuPNvzLQjaOuBwMk555Lc40acSWWaX0ZuaXlldS2/XtjY3NreKe7utWQYC0KbJOSh6LhYUs4C2lRMcdqJBMW+y2nbHdVTv31PhWRhcKvGEXV8PAiYxwhWWuoVS61eYhPM0U19gn74dXo5R+WqHbHjuwudMivmFGiRWBkpQYZGr/hp90MS+zRQhGMpu5YZKSfBQjHC6aRgx5JGmIzwgHY1DbBPpZNMl5mgI630kRcKfQKFpurviQT7Uo59Vyd9rIZy3kvF/7xurLwzJ2FBFCsakNlDXsyRClHaDOozQYniY00wEUz/FZEhFpgo3V9Bl2DNr7xIWtWKZVasq5NSrZrVkYcDOIQyWHAKNbiEBjSBwAM8wQu8Go/Gs/FmvM+iOSOb2Yc/MD6+AUFMmSA=</latexit>

cluster and reciprocal cluster vectors




















reciprocal space





R = eR+ r<latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit><latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit><latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit><latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit>

k = ek + eG<latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit><latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit><latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit><latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit>

Fourier transformations




















reciprocal space





R = eR+ r<latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit><latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit><latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit><latexit sha1_base64="zmNckJX5mdtR07hESM13uYv6RFg=">AAACJnicbVDLSsNAFJ34rPUVdelmsAiCUJIi6KZQcOOyin1AE8pkctMOnTyYmSgl9Gvc+CtuXFRE3PkpTtosbOuBYQ7n3Mu993gJZ1JZ1rextr6xubVd2inv7u0fHJpHx20Zp4JCi8Y8Fl2PSOAsgpZiikM3EUBCj0PHG93mfucJhGRx9KjGCbghGUQsYJQoLfXNeuZ4MfflONQffpjguvPMfFCM+7BsXeIFRUxw36xYVWsGvErsglRQgWbfnDp+TNMQIkU5kbJnW4lyMyIUoxwmZSeVkBA6IgPoaRqREKSbzc6c4HOt+DiIhX6RwjP1b0dGQpnvpitDooZy2cvF/7xeqoIbN2NRkiqI6HxQkHKsYpxnhn0mgCo+1oRQwfSumA6JIFTpZMs6BHv55FXSrlVtq2rfX1UatSKOEjpFZ+gC2egaNdAdaqIWougFvaEp+jBejXfj0/ial64ZRc8JWoDx8wsep6YT</latexit>

k = ek + eG<latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit><latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit><latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit><latexit sha1_base64="3UEaLY/6QxghkCnbLetxTq97SXE=">AAACL3icdVDLSsNAFJ34rPUVdelmsAiCUJIi6EYoCOqygn1AE8pkctMOnTyYmSgl9I/c+CvdiCji1r9w0mahrR4Y5nDOvdx7j5dwJpVlvRpLyyura+uljfLm1vbOrrm335JxKig0acxj0fGIBM4iaCqmOHQSAST0OLS94VXutx9ASBZH92qUgBuSfsQCRonSUs+8zhwv5r4chfrDwzG+dB6ZD4pxH+at03+smzHumRWrak2BF4ldkAoq0OiZE8ePaRpCpCgnUnZtK1FuRoRilMO47KQSEkKHpA9dTSMSgnSz6b1jfKwVHwex0C9SeKr+7MhIKPPddGVI1EDOe7n4l9dNVXDhZixKUgURnQ0KUo5VjPPwsM8EUMVHmhAqmN4V0wERhCodcVmHYM+fvEhataptVe27s0q9VsRRQofoCJ0gG52jOrpFDdREFD2hCXpD78az8WJ8GJ+z0iWj6DlAv2B8fQNuBapr</latexit>

UR,k =

1pLexp(ikR)

<latexit sha1_base64="LpPXl1g8A8mK7aMB/jSuaNhEiRc=">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</latexit><latexit sha1_base64="LpPXl1g8A8mK7aMB/jSuaNhEiRc=">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</latexit><latexit sha1_base64="LpPXl1g8A8mK7aMB/jSuaNhEiRc=">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</latexit><latexit sha1_base64="LpPXl1g8A8mK7aMB/jSuaNhEiRc=">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</latexit>

VeR,ek =

1pL/Lc

exp(iek eR)

<latexit sha1_base64="O7nlfa5iMY8/HMaayJ1J1OYvEew=">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</latexit><latexit sha1_base64="O7nlfa5iMY8/HMaayJ1J1OYvEew=">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</latexit><latexit sha1_base64="O7nlfa5iMY8/HMaayJ1J1OYvEew=">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</latexit><latexit sha1_base64="O7nlfa5iMY8/HMaayJ1J1OYvEew=">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</latexit>

Wr,eG =

1pLc

exp(ieGr)<latexit sha1_base64="6uXd7QXSqiiWms3XQe9M4yHGF6k=">AAACX3icdVFNS8QwEE3r9/pV9SRegougIEu7CHoRBA968KDgusJ2KWk61WD6YTJVl9A/6U3w4j8xu+5BVx0Iebw3j5m8xKUUGn3/zXGnpmdm5+YXGotLyyur3tr6jS4qxaHDC1mo25hpkCKHDgqUcFsqYFksoRs/nA717hMoLYr8Ggcl9DN2l4tUcIaWirynbmRMGBcy0YPMXlTV++GzSACFTOCHclbXjeNGmCrGTVCbUD8qNBcRt3QIL+Wu+NdHJybs0chr+i1/VPQ3CMagScZ1GXmvYVLwKoMcuWRa9wK/xL5hCgWXYDeoNJSMP7A76FmYswx034zyqemOZRKaFsqeHOmI/e4wLNPD5WxnxvBeT2pD8i+tV2F61DciLyuEnH8NSitJsaDDsGkiFHCUAwsYV8LuSvk9swGi/ZKGDSGYfPJvcNNuBX4ruDponrTHccyTLbJNdklADskJOSeXpEM4eXdcZ9FZcj7cOXfF9b5aXWfs2SA/yt38BDEPt6c=</latexit><latexit sha1_base64="6uXd7QXSqiiWms3XQe9M4yHGF6k=">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</latexit><latexit sha1_base64="6uXd7QXSqiiWms3XQe9M4yHGF6k=">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</latexit><latexit sha1_base64="6uXd7QXSqiiWms3XQe9M4yHGF6k=">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</latexit>

lattice F.T. (L x L)

superlattice F.T. (L/Lc x L/Lc)

cluster F.T. (Lc x Lc)

note: U 6= V W = WV<latexit sha1_base64="26eYaA+aPDZNsvjVGU+e+Gm505Y=">AAACQHicbVDPS8MwGE39OeuvqkcvwSF4Gu0Q9CIMvHicYNfBVkaapltYmpQkFUbZn+bFP8GbZy8eFPHqyXTrwW0+CHl57/v4vrwoY1Rp13211tY3Nre2azv27t7+waFzdNxRIpeY+FgwIbsRUoRRTnxNNSPdTBKURowE0fi29INHIhUV/EFPMhKmaMhpQjHSRho4QdGPBIvVJDUX9Kd2nxN7QetMF9/B1L5ZUZY77IFTdxvuDHCVeBWpgwrtgfPSjwXOU8I1ZkipnudmOiyQ1BQzYvbKFckQHqMh6RnKUUpUWMwCmMJzo8QwEdIcruFM/dtRoFSVy5nKFOmRWvZK8T+vl+vkOiwoz3JNOJ4PSnIGtYBlmjCmkmDNJoYgLKnZFeIRkghrk3kZgrf85VXSaTY8t+HdX9ZbzSqOGjgFZ+ACeOAKtMAdaAMfYPAE3sAH+LSerXfry/qel65ZVc8JWID18wvJY6/D</latexit><latexit sha1_base64="26eYaA+aPDZNsvjVGU+e+Gm505Y=">AAACQHicbVDPS8MwGE39OeuvqkcvwSF4Gu0Q9CIMvHicYNfBVkaapltYmpQkFUbZn+bFP8GbZy8eFPHqyXTrwW0+CHl57/v4vrwoY1Rp13211tY3Nre2azv27t7+waFzdNxRIpeY+FgwIbsRUoRRTnxNNSPdTBKURowE0fi29INHIhUV/EFPMhKmaMhpQjHSRho4QdGPBIvVJDUX9Kd2nxN7QetMF9/B1L5ZUZY77IFTdxvuDHCVeBWpgwrtgfPSjwXOU8I1ZkipnudmOiyQ1BQzYvbKFckQHqMh6RnKUUpUWMwCmMJzo8QwEdIcruFM/dtRoFSVy5nKFOmRWvZK8T+vl+vkOiwoz3JNOJ4PSnIGtYBlmjCmkmDNJoYgLKnZFeIRkghrk3kZgrf85VXSaTY8t+HdX9ZbzSqOGjgFZ+ACeOAKtMAdaAMfYPAE3sAH+LSerXfry/qel65ZVc8JWID18wvJY6/D</latexit><latexit sha1_base64="26eYaA+aPDZNsvjVGU+e+Gm505Y=">AAACQHicbVDPS8MwGE39OeuvqkcvwSF4Gu0Q9CIMvHicYNfBVkaapltYmpQkFUbZn+bFP8GbZy8eFPHqyXTrwW0+CHl57/v4vrwoY1Rp13211tY3Nre2azv27t7+waFzdNxRIpeY+FgwIbsRUoRRTnxNNSPdTBKURowE0fi29INHIhUV/EFPMhKmaMhpQjHSRho4QdGPBIvVJDUX9Kd2nxN7QetMF9/B1L5ZUZY77IFTdxvuDHCVeBWpgwrtgfPSjwXOU8I1ZkipnudmOiyQ1BQzYvbKFckQHqMh6RnKUUpUWMwCmMJzo8QwEdIcruFM/dtRoFSVy5nKFOmRWvZK8T+vl+vkOiwoz3JNOJ4PSnIGtYBlmjCmkmDNJoYgLKnZFeIRkghrk3kZgrf85VXSaTY8t+HdX9ZbzSqOGjgFZ+ACeOAKtMAdaAMfYPAE3sAH+LSerXfry/qel65ZVc8JWID18wvJY6/D</latexit><latexit sha1_base64="26eYaA+aPDZNsvjVGU+e+Gm505Y=">AAACQHicbVDPS8MwGE39OeuvqkcvwSF4Gu0Q9CIMvHicYNfBVkaapltYmpQkFUbZn+bFP8GbZy8eFPHqyXTrwW0+CHl57/v4vrwoY1Rp13211tY3Nre2azv27t7+waFzdNxRIpeY+FgwIbsRUoRRTnxNNSPdTBKURowE0fi29INHIhUV/EFPMhKmaMhpQjHSRho4QdGPBIvVJDUX9Kd2nxN7QetMF9/B1L5ZUZY77IFTdxvuDHCVeBWpgwrtgfPSjwXOU8I1ZkipnudmOiyQ1BQzYvbKFckQHqMh6RnKUUpUWMwCmMJzo8QwEdIcruFM/dtRoFSVy5nKFOmRWvZK8T+vl+vkOiwoz3JNOJ4PSnIGtYBlmjCmkmDNJoYgLKnZFeIRkghrk3kZgrf85VXSaTY8t+HdX9ZbzSqOGjgFZ+ACeOAKtMAdaAMfYPAE3sAH+LSerXfry/qel65ZVc8JWID18wvJY6/D</latexit>

overview.6 Michael Potthoff

object, quantity symbol, definition properties, relations

basis spanning the lattice a↵ ↵ = 1, ..., Dvolume of a primitive cell VC = det({a↵})

lattice vectors R = Ri =P

↵ i↵a↵ i↵ 2 Z, 1 i↵ L↵

basis vectors of reciprocal lattice b� a↵b� = 2⇡�↵�

volume of a reciprocal unit cell VRC = det({bj}) VRC = (2⇡)D/VCreciprocal lattice vectors G = Gj =

P� j�b� j� 2 Z, GR 2 2⇡Z

basis spanning the superlattice ea↵ e

a↵ = Lc,↵a↵

volume of a superlattice cell VSC = det({ea↵}) VSC = LcVC

superlattice vectors eR =

eRi =

P↵ i↵e

a↵ i↵ 2 Z, 1 i↵ L/Lc,↵

basis of reciprocal superlattice eb� e

a↵eb� = 2⇡�↵�

volume of a rec. superlattice cell VRSC = det({ebj}) VRSC = (2⇡)D/VSC

reciprocal superlattice vectors eG =

eGj =

P� j�

eb�

eG

eR 2 2⇡Z

vectors spanning the system A↵ A↵ = L↵a↵ = (L/Lc,↵)ea↵

system volume V = det({Ai}) V = LVC. . .

discrete wave vectors k kA↵ 2 2⇡Zvolume element in k-space �Dk �Dk = (2⇡)D/V

. . .

Table 1: Overview of the various quantities defining the lattice and the reciprocal lattice, thesystem volume and the k-space.

series:f(x) =

X

G

fG

eiGx , fG

=

1

VC

Z

CdDrf(x)e�iGx . (5)

On a suitably defined Hilbert space of functions f(x), this expansion results from the fact thatthe functions exp(iGx) for an orthonormal basis: 1

VC

RC d

Dre�iGxeiG0x

= �G,G0 . A completely

analogous construction can be made for functions g(p) with g(p + Gj) = g(p).

While one is eventually interested in the properties of a lattice-fermion model, such as theHubbard model, on a lattice with an infinite number of lattice sites L, it is convenient to firstconsider a finite lattice and to solve the problem for L < 1,

L = L1 ⇥ · · · ⇥ LD , (6)

and a certain finite number of fermions N , and finally to perform the thermodynamical limitL ! 1, N ! 1 with n ⌘ N/L = const. This raises the question of boundary condi-tions. Excluding systems in a critical state where system properties are infinitely susceptibleto small perturbations, the precise form of the boundary conditions does not matter in the ther-modynamic limit. In the case of open boundaries, the system volume is spanned by a set ofvectors A1, ...,AD, such that the system volume is V = det(A1, ...,AD). We should imposeconsistency constraints, A↵ = L↵a↵, to respect the underlying lattice structure.

GREEN’S FUNCTION

single-electron Green’s function

Cluster Extensions of Dynamical Mean-Field Theory .9

transformation (U ), the superlattice Fourier transformation (V ), and the cluster Fourier trans-formation (W ). For example, one can straightforwardly prove that

1

L

X

R

e�i(k�k

0)R=

X

R

U †k,RUR,k0

= �k,k0 ,

1

L

RCX

k

eik(R�R

0)=

X

k

UR,kU

†k,R0 = �

R,R0 (14)

and thus, for some R-dependent observable OR

,

Ok

=

1pL

X

R

eikROR

, OR

=

1pL

X

k

e�ikROk

. (15)

Analogous relations hold for the other two cases. The formal structure of the proof is the samein all cases. Although there is a one-to-one relation between R and (

eR, r) and between k and

(

ek, e

G), the lattice Fourier transformation U involving lattice vectors R cannot be decomposedinto the superlattice transformation involving superlattice vectors e

R and the cluster transforma-tion involving cluster vectors r, i.e., U 6= V W = WV . We note that a quantity A

R,R0 whichis invariant under lattice translations R0, i.e. A

R+R0,R0+R0 = AR,R0 , is diagonalized by U :

(U

†AU )

kk

0= A(k)�

k,k0 . A quantity AR,R0 which is invariant under superlattice translations

eR0 as well as under cluster translations r0 (i.e. which is cyclic on the cluster), A

R+ eR0,R0+ e

R0=

AR+r0,R0+r0 = A

R,R0 , is diagonalized by V W : (W †V

†AV W )e

k

eG,ek0f

G

0 = A(ek, eG)�e

k,ek0� eG, eG0 .

4 Single-electron Green’s function

DMFT and its cluster extensions are theories based on the single-particle Green’s functionG

R,R0(!) which for complex ! 6= R (and using units with ~ = 1) is defined as

GR,R0

(!) = h0|c†R

0,�

1

! � E0 +HcR,�|0i + h0|c

R,�1

! + E0 � Hc†R

0,�|0i . (16)

Here, c†R,� is the creation operator of an electron with spin projection � =", # at the lattice site

R, and cR,� is the corresponding annihilator. |0i is the ground state of the Hamiltonian H , and

E0 is the ground-state energy. Note that 1/(...) stands for the operator inverse. We assume thatH is spin-independent, such that the Green’s function is diagonal in the spin-projection indicesand spin-independent. Furthermore, H shall be invariant under lattice translations, c

R,� !cR+R0,� and c†

R,� ! c†R+R0,�. Then, the Green’s function is invariant under lattice translations

as well,G

R+R0,R0+R0(!) = GR,R0

(!) . (17)

This implies that the matrix G(!) with elements GR,R0

(!) is diagonalized by the lattice Fouriertransformation:

(U

†G(!)U )

kk

0= G(k,!)�

k,k0 . (18)

From the retarded Green’s function G

(ret)(!) ⌘ lim⌘&0 G(! + i⌘) ⌘ G(! + i0+

) with ! 2 Rwe obtain the spectral density

A(k,!) = � 1

⇡ImG(k,! + i0+

) . (19)



1

L

X

R

e�i(k�k

0)R=

X

R

U †k,RUR,k0

= �k,k0 ,

1

L

RCX

k

eik(R�R

0)=

X

k

UR,kU

†k,R0 = �

R,R0 (14)


,

Ok

=

1pL

X

R

eikROR

, OR

=

1pL

X

k

e�ikROk

. (15)



(

ek, e





(U

†AU )

kk

0= A(k)�



R+ eR0,R0+ e

R0=

AR+r0,R0+r0 = A


†AV W )e

k

eG,ek0f

G

0 = A(ek, eG)�e

k,ek0� eG, eG0 .




GR,R0

(!) = h0|c†R

0,�

1

! � E0 +HcR,�|0i + h0|c

R,�1

! + E0 � Hc†R

0,�|0i . (16)






as well,G

R+R0,R0+R0(!) = GR,R0

(!) . (17)



(U

†G(!)U)

kk

0= G(k,!)�

k,k0 . (18)


(ret)(!) ⌘ lim⌘&0 G(! + i⌘) ⌘ G(! + i0+


A(k,!) = � 1

⇡ImG(k,! + i0+

) . (19)



1

L

X

R

e�i(k�k

0)R=

X

R

U †k,RUR,k0

= �k,k0 ,

1

L

RCX

k

eik(R�R

0)=

X

k

UR,kU

†k,R0 = �

R,R0 (14)


,

Ok

=

1pL

X

R

eikROR

, OR

=

1pL

X

k

e�ikROk

. (15)



(

ek, e





(U

†AU )

kk

0= A(k)�



R+ eR0,R0+ e

R0=

AR+r0,R0+r0 = A


†AV W )e

k

eG,ek0f

G

0 = A(ek, eG)�e

k,ek0� eG, eG0 .




GR,R0

(!) = h0|c†R

0,�

1

! � E0 +HcR,�|0i + h0|c

R,�1

! + E0 � Hc†R

0,�|0i . (16)






as well,G

R+R0,R0+R0(!) = GR,R0

(!) . (17)



(U

†G(!)U)

kk

0= G(k,!)�

k,k0 . (18)


(ret)(!) ⌘ lim⌘&0 G(! + i⌘) ⌘ G(! + i0+


A(k,!) = � 1

⇡ImG(k,! + i0+

) . (19)



1

L

X

R

e�i(k�k

0)R=

X

R

U †k,RUR,k0

= �k,k0 ,

1

L

RCX

k

eik(R�R

0)=

X

k

UR,kU

†k,R0 = �

R,R0 (14)


,

Ok

=

1pL

X

R

eikROR

, OR

=

1pL

X

k

e�ikROk

. (15)



(

ek, e





(U

†AU )

kk

0= A(k)�



R+ eR0,R0+ e

R0=

AR+r0,R0+r0 = A


†AV W )e

k

eG,ek0f

G

0 = A(ek, eG)�e

k,ek0� eG, eG0 .




GR,R0

(!) = h0|c†R

0,�

1

! � E0 +HcR,�|0i + h0|c

R,�1

! + E0 � Hc†R

0,�|0i . (16)






as well,G

R+R0,R0+R0(!) = GR,R0

(!) . (17)



(U

†G(!)U)

kk

0= G(k,!)�

k,k0 . (18)


(ret)(!) ⌘ lim⌘&0 G(! + i⌘) ⌘ G(! + i0+


A(k,!) = � 1

⇡ImG(k,! + i0+

) . (19)

single-electron Green’s function (at temperature T=0)

H invariant under lattice translations:cR,� ! cR+R0,� c†R,� ! c†R+R0,�

<latexit sha1_base64="efymHiGNgMSnKhVvbFOcqhHRt/o=">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</latexit><latexit sha1_base64="efymHiGNgMSnKhVvbFOcqhHRt/o=">AAACjnicdVHbSgMxEM2ut9p6WfXRl2ARBKXsFlFRxIIvfaxiq9Cty2w23YZmL02yQln6Of6Qb/6N6UWwrR0IOTlzZjI58VPOpLLtb8NcW9/Y3CpsF0s7u3v71sFhSyaZILRJEp6INx8k5SymTcUUp2+poBD5nL76/cdx/vWDCsmS+EUNU9qJIIxZlxFQmvKsT+Llrp/wQA4jveHnC1eyMIIRdlWCF5P4HM+dPftXXnQHgwyCInl3AwhDKla0nXRdoVnd3bPKdsWeBF4GzgyU0SwanvXlBgnJIhorwkHKtmOnqpODUIxwqqfNJE2B9CGkbQ1jiKjs5BM7R/hUMwHuJkKvWOEJ+7cih0iOZ9TKCFRPLubG5H+5dqa6N52cxWmmaEymF3UzjrUn47/BAROUKD7UAIhgelZMeiCAKP2DRW2Cs/jkZdCqVhy74jxdlmvVmR0FdIxO0Bly0DWqoTpqoCYiRslwjFvjzrTMK/PefJhKTWNWc4Tmwqz/ALEhxyY=</latexit><latexit sha1_base64="efymHiGNgMSnKhVvbFOcqhHRt/o=">AAACjnicdVHbSgMxEM2ut9p6WfXRl2ARBKXsFlFRxIIvfaxiq9Cty2w23YZmL02yQln6Of6Qb/6N6UWwrR0IOTlzZjI58VPOpLLtb8NcW9/Y3CpsF0s7u3v71sFhSyaZILRJEp6INx8k5SymTcUUp2+poBD5nL76/cdx/vWDCsmS+EUNU9qJIIxZlxFQmvKsT+Llrp/wQA4jveHnC1eyMIIRdlWCF5P4HM+dPftXXnQHgwyCInl3AwhDKla0nXRdoVnd3bPKdsWeBF4GzgyU0SwanvXlBgnJIhorwkHKtmOnqpODUIxwqqfNJE2B9CGkbQ1jiKjs5BM7R/hUMwHuJkKvWOEJ+7cih0iOZ9TKCFRPLubG5H+5dqa6N52cxWmmaEymF3UzjrUn47/BAROUKD7UAIhgelZMeiCAKP2DRW2Cs/jkZdCqVhy74jxdlmvVmR0FdIxO0Bly0DWqoTpqoCYiRslwjFvjzrTMK/PefJhKTWNWc4Tmwqz/ALEhxyY=</latexit><latexit sha1_base64="efymHiGNgMSnKhVvbFOcqhHRt/o=">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</latexit>

implies:

hence: G can be diagonalized using U

spectral density: (the central observable, related to PES, IPE)

noninteracting spectral density

0

42

−4−6−8

ΓMXΓk

8

−2

6E!<latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit><latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit><latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit><latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit>

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT

!<latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit><latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit><latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit><latexit sha1_base64="lG30MxnOwwtH5vtSgE16OQC/CzY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMeAF48RzAOSJcxOOsmQmdllZlYISz7CiwdFvPo93vwbZ5M9aGJBQ1HVTXdXlAhurO9/e6WNza3tnfJuZW//4PCoenzSNnGqGbZYLGLdjahBwRW2LLcCu4lGKiOBnWh6l/udJ9SGx+rRzhIMJR0rPuKMWid1+rHEMa0MqjW/7i9A1klQkBoUaA6qX/1hzFKJyjJBjekFfmLDjGrLmcB5pZ8aTCib0jH2HFVUogmzxblzcuGUIRnF2pWyZKH+nsioNGYmI9cpqZ2YVS8X//N6qR3dhhlXSWpRseWiUSqIjUn+OxlyjcyKmSOUae5uJWxCNWXWJZSHEKy+vE7aV/XArwcP17XGdRFHGc7gHC4hgBtowD00oQUMpvAMr/DmJd6L9+59LFtLXjFzCn/gff4AwnaPIQ==</latexit>

.10 Michael Potthoff

This is a central observable which is related, for ! < 0, to the k-resolved (angle-resolved)photoemission spectrum (PES), and, for ! > 0, to the inverse photoemission spectrum (inversephotoemission, IPE). PES and IPE probe the occupied and the unoccupied part of the electronicstructure in the vicinity of the Fermi energy (! = 0), respectively. With the help of the Diracidentity, lim⌘&0 1/(x+ i⌘) = P(1/x) � i⇡�(x), we also have the expression

AR,R0

(!) = h0|c†R

0,��(! � E0 +H)cR,�|0i + h0|c

R,��(! + E0 � H)c†R

0,�|0i (20)

for real !, and

GR,R0

(!) =

Z 1

�1d!0AR,R0

(!0)

! � !0 . (21)

The simplest case is that of a noninteracting tight-binding model with Hamiltonian

H0 =

X

RR

0�

tRR

0c†R�cR0� . (22)

Expressing the �-function in Eq. (20) as �(!) = (2⇡)�1Rdt e�i!t, using the Baker-Campbell-

Hausdorff formula, eBAe�B= e�LBA for operators A,B and LBA ⌘ [A,B], and the elemen-

tary commutator [cR�, H0] =

PR

0 tR,R0c

R

0�, we find

A(!) = �(! � t) , A(k,!) = �(! � ✏(k)) , (23)

where ✏(k) are the eigenvalues of the hopping matrix t, i.e., the tight-binding dispersion. Here,we discuss the case of electrons on the two-dimensional square lattice with hopping betweennearest neighbors only, i.e., t

R,R0= �t if R, R

0 are nearest-neighbor sites. t = 1 fixesthe energy scale. The dispersion is obtained by lattice Fourier transformation, "(k)�

kk

0=

(U

†tU )

kk

0 :"(k) = �2t(cos kx + cos ky) . (24)

With this we obtain the noninteracting spectral density as displayed in Fig. 3 (left). A(k,!)

has exactly one �-singularity for each k in the reciprocal unit cell, e.g., in the first Brillouinzone. There is a well-defined Fermi surface, given by all k-points in the BZ with vanishingexcitation energy: ! = ✏(k) = 0. The band is half-filled. Different band fillings are obtainedby introducing a chemical potential µ 6= 0 via the substitution H0 7! H0 � µN , where N is theparticle-number operator.

The interacting spectral density Fig. 3 (right) looks quite different. The example shows A(k,!)for the same tight-binding model but with an additional on-site Coulomb interaction

H1 =U

2

X

R�

nR�nR�� (25)

of strength U , where nR� ⌘ c†

R�cR� is the occupation-number operator. H = H0 + H1 isthe famous Hubbard model [18–20] which frequently serves as a prototype for method devel-opments. The computation of the spectral density in the interacting cases is, of course, by no



AR,R0

(!) = h0|c†R

0,��(! � E0 +H)cR,�|0i + h0|c

R,��(! + E0 � H)c†R

0,�|0i (20)

for real !, and

GR,R0

(!) =

Z 1

�1d!0AR,R0

(!0)

! � !0 . (21)


H0 =

X

RR

0�

tRR

0c†R�cR0� . (22)




PR

0 tR,R0c

R

0�, we find

A(!) = �(! � t) , A(k,!) = �(! � ✏(k)) , (23)


R,R0= �t if R, R


kk

0=

(U

†tU )

kk

0 :"(k) = �2t(cos kx + cos ky) . (24)




H1 =U

2

X

R�

nR�nR�� (25)





AR,R0

(!) = h0|c†R

0,��(! � E0 +H)cR,�|0i + h0|c

R,��(! + E0 � H)c†R

0,�|0i (20)

for real !, and

GR,R0

(!) =

Z 1

�1d!0AR,R0

(!0)

! � !0 . (21)


H0 =

X

RR

0�

tRR

0c†R�cR0� . (22)




PR

0 tR,R0c

R

0�, we find

A(!) = �(! � t) , A(k,!) = �(! � ✏(k)) , (23)


R,R0= �t if R, R


kk

0=

(U

†tU )

kk

0 :"(k) = �2t(cos kx + cos ky) . (24)




H1 =U

2

X

R�

nR�nR�� (25)





AR,R0

(!) = h0|c†R

0,��(! � E0 +H)cR,�|0i + h0|c

R,��(! + E0 � H)c†R

0,�|0i (20)

for real !, and

GR,R0

(!) =

Z 1

�1d!0AR,R0

(!0)

! � !0 . (21)


H0 =

X

RR

0�

tRR

0c†R�cR0� . (22)




PR

0 tR,R0c

R

0�, we find

A(!) = �(! � t) , A(k,!) = �(! � ✏(k)) , (23)


R,R0= �t if R, R


kk

0=

(U

†tU )

kk

0 :"(k) = �2t(cos kx + cos ky) . (24)




H1 =U

2

X

R�

nR�nR�� (25)



• half-filling, else:!• occupied states - PES!• unoccupied states - IPE!• Fermi surface !

What happens when adding an interaction? (e.g. single-band Hubbard model)!

H ! H � µN<latexit sha1_base64="N4RGwEydcop0e5JzOpEsiFyqZMk=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBi2W3CHoseOlJKtgP6C4lm2bb0CS7JtlCWfo7vHhQxKs/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKG5tb2zvF3dLe/sHhUfn4pK3jVBHaIjGPVTfEmnImacsww2k3URSLkNNOOL6b+50JVZrF8tFMExoIPJQsYgQbKwUN38Soga58kaL7frniVt0F0DrxclKBHM1++csfxCQVVBrCsdY9z01MkGFlGOF0VvJTTRNMxnhIe5ZKLKgOssXRM3RhlQGKYmVLGrRQf09kWGg9FaHtFNiM9Ko3F//zeqmJboOMySQ1VJLloijlyH46TwANmKLE8KklmChmb0VkhBUmxuZUsiF4qy+vk3at6rlV7+G6Uq/lcRThDM7hEjy4gTo0oAktIPAEz/AKb87EeXHenY9la8HJZ07hD5zPH+rjkNQ=</latexit><latexit sha1_base64="N4RGwEydcop0e5JzOpEsiFyqZMk=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBi2W3CHoseOlJKtgP6C4lm2bb0CS7JtlCWfo7vHhQxKs/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKG5tb2zvF3dLe/sHhUfn4pK3jVBHaIjGPVTfEmnImacsww2k3URSLkNNOOL6b+50JVZrF8tFMExoIPJQsYgQbKwUN38Soga58kaL7frniVt0F0DrxclKBHM1++csfxCQVVBrCsdY9z01MkGFlGOF0VvJTTRNMxnhIe5ZKLKgOssXRM3RhlQGKYmVLGrRQf09kWGg9FaHtFNiM9Ko3F//zeqmJboOMySQ1VJLloijlyH46TwANmKLE8KklmChmb0VkhBUmxuZUsiF4qy+vk3at6rlV7+G6Uq/lcRThDM7hEjy4gTo0oAktIPAEz/AKb87EeXHenY9la8HJZ07hD5zPH+rjkNQ=</latexit><latexit sha1_base64="N4RGwEydcop0e5JzOpEsiFyqZMk=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBi2W3CHoseOlJKtgP6C4lm2bb0CS7JtlCWfo7vHhQxKs/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKG5tb2zvF3dLe/sHhUfn4pK3jVBHaIjGPVTfEmnImacsww2k3URSLkNNOOL6b+50JVZrF8tFMExoIPJQsYgQbKwUN38Soga58kaL7frniVt0F0DrxclKBHM1++csfxCQVVBrCsdY9z01MkGFlGOF0VvJTTRNMxnhIe5ZKLKgOssXRM3RhlQGKYmVLGrRQf09kWGg9FaHtFNiM9Ko3F//zeqmJboOMySQ1VJLloijlyH46TwANmKLE8KklmChmb0VkhBUmxuZUsiF4qy+vk3at6rlV7+G6Uq/lcRThDM7hEjy4gTo0oAktIPAEz/AKb87EeXHenY9la8HJZ07hD5zPH+rjkNQ=</latexit><latexit sha1_base64="N4RGwEydcop0e5JzOpEsiFyqZMk=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBi2W3CHoseOlJKtgP6C4lm2bb0CS7JtlCWfo7vHhQxKs/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKG5tb2zvF3dLe/sHhUfn4pK3jVBHaIjGPVTfEmnImacsww2k3URSLkNNOOL6b+50JVZrF8tFMExoIPJQsYgQbKwUN38Soga58kaL7frniVt0F0DrxclKBHM1++csfxCQVVBrCsdY9z01MkGFlGOF0VvJTTRNMxnhIe5ZKLKgOssXRM3RhlQGKYmVLGrRQf09kWGg9FaHtFNiM9Ko3F//zeqmJboOMySQ1VJLloijlyH46TwANmKLE8KklmChmb0VkhBUmxuZUsiF4qy+vk3at6rlV7+G6Uq/lcRThDM7hEjy4gTo0oAktIPAEz/AKb87EeXHenY9la8HJZ07hD5zPH+rjkNQ=</latexit>

interacting spectral density

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


• 2D Hubbard model, n=1, low T=0.1!• U=W=8!• L=8x8 sites, PBC!• QMC results

Gröber et al. (2000)

interacting spectral density

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT



technical problems:!• k resolution!• thermal broadening!• thermally induced decay of correlations!• MaxEnt must be used to get real-frequency data: broadening


many-body effects

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT



many-body effects: !• gapped single-particle excitations -> interaction-driven (Mott) insulator!• finite lifetime of excitations!• incoherent background: complicated high-order decay processes


Hubbard bands

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


many-body effects: !• Hubbard bands: correlation-induced satellites!• LHB: PES!• LOCAL CORRELATIONS

IPE

U

PESGröber et al. (2000)




AR,R0

(!) = h0|c†R

0,��(! � E0 +H)cR,�|0i + h0|c

R,��(! + E0 � H)c†R

0,�|0i (20)

for real !, and

GR,R0

(!) =

Z 1

�1d!0AR,R0

(!0)

! � !0 . (21)


H0 =

X

RR

0�

tRR

0c†R�cR0� . (22)




PR

0 tR,R0c

R

0�, we find

A(!) = �(! � t) , A(k,!) = �(! � ✏(k)) , (23)


R,R0= �t if R, R


kk

0=

(U

†tU )

kk

0 :"(k) = �2t(cos kx + cos ky) . (24)




H1 =U

2

X

R�

nR�nR�� (25)



! = E0 � En � µ ⇡ �µ = �U/2<latexit sha1_base64="8yHbwsYm2gFgauKOatOnJBfMvaE=">AAACFHicbZDLSgMxFIYz9VbrrerSTbAIgrTOFEE3QkEElxWcttAZSiZN29BchiQjlqEP4cZXceNCEbcu3Pk2pu0stPVAwsf/n0Ny/ihmVBvX/XZyS8srq2v59cLG5tb2TnF3r6FlojDxsWRStSKkCaOC+IYaRlqxIohHjDSj4dXEb94TpakUd2YUk5CjvqA9ipGxUqd4EkhO+ghewuuOC8v2FrAc8CRAcazkw5StWYb+abVTLLkVd1pwEbwMSiCreqf4FXQlTjgRBjOkddtzYxOmSBmKGRkXgkSTGOEh6pO2RYE40WE6XWoMj6zShT2p7BEGTtXfEyniWo94ZDs5MgM9703E/7x2YnoXYUpFnBgi8OyhXsKgkXCSEOxSRbBhIwsIK2r/CvEAKYSNzbFgQ/DmV16ERrXiuRXv9qxUq2Zx5MEBOATHwAPnoAZuQB34AINH8AxewZvz5Lw4787HrDXnZDP74E85nz8me5sK</latexit><latexit sha1_base64="8yHbwsYm2gFgauKOatOnJBfMvaE=">AAACFHicbZDLSgMxFIYz9VbrrerSTbAIgrTOFEE3QkEElxWcttAZSiZN29BchiQjlqEP4cZXceNCEbcu3Pk2pu0stPVAwsf/n0Ny/ihmVBvX/XZyS8srq2v59cLG5tb2TnF3r6FlojDxsWRStSKkCaOC+IYaRlqxIohHjDSj4dXEb94TpakUd2YUk5CjvqA9ipGxUqd4EkhO+ghewuuOC8v2FrAc8CRAcazkw5StWYb+abVTLLkVd1pwEbwMSiCreqf4FXQlTjgRBjOkddtzYxOmSBmKGRkXgkSTGOEh6pO2RYE40WE6XWoMj6zShT2p7BEGTtXfEyniWo94ZDs5MgM9703E/7x2YnoXYUpFnBgi8OyhXsKgkXCSEOxSRbBhIwsIK2r/CvEAKYSNzbFgQ/DmV16ERrXiuRXv9qxUq2Zx5MEBOATHwAPnoAZuQB34AINH8AxewZvz5Lw4787HrDXnZDP74E85nz8me5sK</latexit><latexit sha1_base64="8yHbwsYm2gFgauKOatOnJBfMvaE=">AAACFHicbZDLSgMxFIYz9VbrrerSTbAIgrTOFEE3QkEElxWcttAZSiZN29BchiQjlqEP4cZXceNCEbcu3Pk2pu0stPVAwsf/n0Ny/ihmVBvX/XZyS8srq2v59cLG5tb2TnF3r6FlojDxsWRStSKkCaOC+IYaRlqxIohHjDSj4dXEb94TpakUd2YUk5CjvqA9ipGxUqd4EkhO+ghewuuOC8v2FrAc8CRAcazkw5StWYb+abVTLLkVd1pwEbwMSiCreqf4FXQlTjgRBjOkddtzYxOmSBmKGRkXgkSTGOEh6pO2RYE40WE6XWoMj6zShT2p7BEGTtXfEyniWo94ZDs5MgM9703E/7x2YnoXYUpFnBgi8OyhXsKgkXCSEOxSRbBhIwsIK2r/CvEAKYSNzbFgQ/DmV16ERrXiuRXv9qxUq2Zx5MEBOATHwAPnoAZuQB34AINH8AxewZvz5Lw4787HrDXnZDP74E85nz8me5sK</latexit><latexit sha1_base64="8yHbwsYm2gFgauKOatOnJBfMvaE=">AAACFHicbZDLSgMxFIYz9VbrrerSTbAIgrTOFEE3QkEElxWcttAZSiZN29BchiQjlqEP4cZXceNCEbcu3Pk2pu0stPVAwsf/n0Ny/ihmVBvX/XZyS8srq2v59cLG5tb2TnF3r6FlojDxsWRStSKkCaOC+IYaRlqxIohHjDSj4dXEb94TpakUd2YUk5CjvqA9ipGxUqd4EkhO+ghewuuOC8v2FrAc8CRAcazkw5StWYb+abVTLLkVd1pwEbwMSiCreqf4FXQlTjgRBjOkddtzYxOmSBmKGRkXgkSTGOEh6pO2RYE40WE6XWoMj6zShT2p7BEGTtXfEyniWo94ZDs5MgM9703E/7x2YnoXYUpFnBgi8OyhXsKgkXCSEOxSRbBhIwsIK2r/CvEAKYSNzbFgQ/DmV16ERrXiuRXv9qxUq2Zx5MEBOATHwAPnoAZuQB34AINH8AxewZvz5Lw4787HrDXnZDP74E85nz8me5sK</latexit>

Hubbard bands

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


many-body effects: !• Hubbard bands: correlation-induced satellites!• UHB: IPE!• LOCAL CORRELATIONS

IPE

U

PES



AR,R0

(!) = h0|c†R

0,��(! � E0 +H)cR,�|0i + h0|c

R,��(! + E0 � H)c†R

0,�|0i (20)

for real !, and

GR,R0

(!) =

Z 1

�1d!0AR,R0

(!0)

! � !0 . (21)


H0 =

X

RR

0�

tRR

0c†R�cR0� . (22)




PR

0 tR,R0c

R

0�, we find

A(!) = �(! � t) , A(k,!) = �(! � ✏(k)) , (23)


R,R0= �t if R, R


kk

0=

(U

†tU )

kk

0 :"(k) = �2t(cos kx + cos ky) . (24)




H1 =U

2

X

R�

nR�nR�� (25)



! = En � E0 � µ ⇡ U � µ = U/2<latexit sha1_base64="nKo15aRuhdSnCA9FPsykW/wAM4I=">AAACFXicbVDLSgMxFM3UV62vUZdugkVwYetMEXRTKIjgsoLTFjqlZNK0DeYxJBmxlP6EG3/FjQtF3Aru/BvT6Sy09UDCyTn3cnNPFDOqjed9O7ml5ZXVtfx6YWNza3vH3d1raJkoTAIsmVStCGnCqCCBoYaRVqwI4hEjzejucuo374nSVIpbM4pJh6OBoH2KkbFS1z0JJScDBKvwqitgyd4eLIU8gSGKYyUfYJC+qhAGp5WuW/TKXgq4SPyMFEGGetf9CnsSJ5wIgxnSuu17semMkTIUMzIphIkmMcJ3aEDalgrEie6M060m8MgqPdiXyh5hYKr+7hgjrvWIR7aSIzPU895U/M9rJ6Z/0RlTESeGCDwb1E8YNBJOI4I9qgg2bGQJworav0I8RAphY4Ms2BD8+ZUXSaNS9r2yf3NWrFWyOPLgAByCY+CDc1AD16AOAoDBI3gGr+DNeXJenHfnY1aac7KeffAHzucPxLGbXA==</latexit><latexit sha1_base64="nKo15aRuhdSnCA9FPsykW/wAM4I=">AAACFXicbVDLSgMxFM3UV62vUZdugkVwYetMEXRTKIjgsoLTFjqlZNK0DeYxJBmxlP6EG3/FjQtF3Aru/BvT6Sy09UDCyTn3cnNPFDOqjed9O7ml5ZXVtfx6YWNza3vH3d1raJkoTAIsmVStCGnCqCCBoYaRVqwI4hEjzejucuo374nSVIpbM4pJh6OBoH2KkbFS1z0JJScDBKvwqitgyd4eLIU8gSGKYyUfYJC+qhAGp5WuW/TKXgq4SPyMFEGGetf9CnsSJ5wIgxnSuu17semMkTIUMzIphIkmMcJ3aEDalgrEie6M060m8MgqPdiXyh5hYKr+7hgjrvWIR7aSIzPU895U/M9rJ6Z/0RlTESeGCDwb1E8YNBJOI4I9qgg2bGQJworav0I8RAphY4Ms2BD8+ZUXSaNS9r2yf3NWrFWyOPLgAByCY+CDc1AD16AOAoDBI3gGr+DNeXJenHfnY1aac7KeffAHzucPxLGbXA==</latexit><latexit sha1_base64="nKo15aRuhdSnCA9FPsykW/wAM4I=">AAACFXicbVDLSgMxFM3UV62vUZdugkVwYetMEXRTKIjgsoLTFjqlZNK0DeYxJBmxlP6EG3/FjQtF3Aru/BvT6Sy09UDCyTn3cnNPFDOqjed9O7ml5ZXVtfx6YWNza3vH3d1raJkoTAIsmVStCGnCqCCBoYaRVqwI4hEjzejucuo374nSVIpbM4pJh6OBoH2KkbFS1z0JJScDBKvwqitgyd4eLIU8gSGKYyUfYJC+qhAGp5WuW/TKXgq4SPyMFEGGetf9CnsSJ5wIgxnSuu17semMkTIUMzIphIkmMcJ3aEDalgrEie6M060m8MgqPdiXyh5hYKr+7hgjrvWIR7aSIzPU895U/M9rJ6Z/0RlTESeGCDwb1E8YNBJOI4I9qgg2bGQJworav0I8RAphY4Ms2BD8+ZUXSaNS9r2yf3NWrFWyOPLgAByCY+CDc1AD16AOAoDBI3gGr+DNeXJenHfnY1aac7KeffAHzucPxLGbXA==</latexit><latexit sha1_base64="nKo15aRuhdSnCA9FPsykW/wAM4I=">AAACFXicbVDLSgMxFM3UV62vUZdugkVwYetMEXRTKIjgsoLTFjqlZNK0DeYxJBmxlP6EG3/FjQtF3Aru/BvT6Sy09UDCyTn3cnNPFDOqjed9O7ml5ZXVtfx6YWNza3vH3d1raJkoTAIsmVStCGnCqCCBoYaRVqwI4hEjzejucuo374nSVIpbM4pJh6OBoH2KkbFS1z0JJScDBKvwqitgyd4eLIU8gSGKYyUfYJC+qhAGp5WuW/TKXgq4SPyMFEGGetf9CnsSJ5wIgxnSuu17semMkTIUMzIphIkmMcJ3aEDalgrEie6M060m8MgqPdiXyh5hYKr+7hgjrvWIR7aSIzPU895U/M9rJ6Z/0RlTESeGCDwb1E8YNBJOI4I9qgg2bGQJworav0I8RAphY4Ms2BD8+ZUXSaNS9r2yf3NWrFWyOPLgAByCY+CDc1AD16AOAoDBI3gGr+DNeXJenHfnY1aac7KeffAHzucPxLGbXA==</latexit>

IPE

U

PES



low-energy excitations

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


many-body effects: !• well-defined low-energy structure !• infinite U: fully localized electrons - 2L-fold degenerate ground state!• finite U: delocalization via second-order hopping processes

mapping onto AF Heisenberg model with J=4t2/U!• low-energy excitations: nonlocal spin excitations, band width 2J


Gröber et al. (2000)superexchange

Utt

low-energy excitations

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


many-body effects: !• low-energy excitations: nonlocal spin excitations, band width 2J!• spin excitations couple to single-particle excitations:

moving hole (PES) or doublon (IPE) dressed, emit/absorb spin excitations!• renormalization of the Hubbard bands AND new satellites!• NONLOCAL CORRELATIONS


Gröber et al. (2000)superexchange

Utt

local and nonlocal correlations

LOCAL CORRELATIONS!

• high-energy spectral features!• high spectral weight!• Hubbard bands!

• accessible within mean-field theories with a proper treatment of the local excitations and their delocalization in the lattice!

• DMFT

NONLOCAL CORRELATIONS!

• low-energy spectral features!• lower spectral weight!• hole/doublon after emission/

absorption of a spin excitation!

• requires feedback of two-particle (e.g. AF magnetic) correlations to the single-particle excitations!

• beyond mean-field theory!• cluster extensions of DMFT!

CLUSTER-PERTURBATION THEORY

bad news

= +


1 Motivation

Self-energy-functional theory (SFT) [1–4] is a general theoretical frame that can be used toconstruct various approximate approaches by which the thermal properties and the spectrum ofone-particle excitations of a certain class of correlated electron systems can be studied. Theprototype considered here is the single-band Hubbard model [5–7] but, quite generally, the SFTapplies to models of strongly correlated fermions on three or lower-dimensional lattices withlocal interactions.There are several extensions of the theory, e.g. to systems with non-local interactions [8], tobosonic systems [9, 10] and the Jaynes-Cummings lattice [11, 12], to electron-phonon systems[13], to systems with quenched disorder [14] as well as for the study of the real-time dynamicsof systems far from thermal equilibrium [15]. To be concise, those extensions will not becovered here.The prime example of an approximation that can be constructed within the SFT is the variationalcluster approximation (VCA) [2, 16]. Roughly, one of the main ideas of the VCA is to adopt a“divide and conquer strategy”: A tiling of the original lattice into disconnected small clustersis considered, as shown in Fig. 1 for example. While the Hubbard model on the infinite squarelattice cannot be solved exactly, there are no serious practical problems to the solve the samemodel for an isolated cluster or for a set of disconnected clusters. The VCA constructs anapproximate solution for the infinite lattice from the solution of the individual clusters by meansof all-order perturbation theory for those terms in the Hamiltonian that connect the clusters.This is actually the concept of the so-called cluster perturbation theory (CPT) [17, 18]. How-ever, it is not sufficient in most cases, and we would like to go beyond the CPT. The essentialproblem becomes apparent e.g. for a system with spontaneously broken symmetry such as anantiferromagnet. The antiferromagnetic state is characterized by a finite value for the sublatticemagnetization which serves as an order parameter. On the other hand, quite generally, the orderparameter must be zero for a system of finite size and thus for a small cluster in particular. Cou-pling finite (and thus necessarily non-magnetic) clusters by means of the CPT, however, onenever gets to an antiferromagnetic solution for the infinite lattice. “Divide and conquer” is not

= +

Fig. 1: Sketch of the decomposition of the original system H = H0

(t) + H1

into a referencesystem H 0

= H0

(t0) +H1

and the inter-cluster hopping H0

(V ) for a square lattice and clustersize L

c

= 16. Blue lines: nearest-neighbor hopping t. Red dots: on-site Hubbard interaction.

• strong finite-size artefacts!• all excitations gapped!• no phase transitions

no phase diagrams

:-(

phase diagram of high-TC materials

the high-TC problem, 10 years ago

From the microscopic physics of a cluster to the macroscopic world of solids

TP I

Relevante Fragen

1. Was ist der Mechanismus der Paarbildung?Antiferromagnetismus

2. Wie bildet sich der makroskopischesupraleitende Zustand?

?*T

Tc

What we wereable to do ...

... where we wanted to go ...

!!x" becomes a straight line between the two boundaries ofthe phase-separation region x1 and x2.24 Whether the exactground state supports phase separation can be derived fromthe finite-size scaling of the doping discontinuity "x#x2−x1, see Table I. Unfortunately, no regular finite-size behav-ior can be inferred from Fig. 2 and Table I for hole doping,probably due to the fact that the clusters are still too small.Opposed to the clear trend visible for the electron-doped case!see Table I", there is a much weaker Lc dependence of thediscontinuities "x, "M, and "D, which we rather interpretas being irregular. However, our results do not exclude mi-croscopic phase separation to persist for Lc→#. The inclu-sion of long-range Coulomb interaction would then be nec-essary in order to “frustrate” the phase separation occurringin the plain Hubbard model and produce microscopic inho-mogeneous phases, such as stripes.23,27,28 We stress that atthis point only qualitative estimates for Lc→# rather than aconvincing finite-size scaling are possible. For a discussionon these issues see, e.g., Refs. 24–26.

The situation is quite different in the electron-doped case!see Fig. 3". Here, not only the phase-separation energy "!,

but also the doping discontinuity "x appears to vanish forLc→#. In fact, "! is already an order of magnitude smallerthan for hole doping in the Lc=4 cluster.8 In addition, alreadyfor Lc=10 the transition from the AF+SC to the pure SCphase has become continuous at least within numerical accu-racy. In this case, the weak phase separation observed at themean-field level for small clusters was simply a signal of atendency of the system to produce microscopically inhomo-geneous phases !such as stripes", as conjectured in Ref. 8.The fact that the corresponding energy scale is already verysmall for a small cluster could explain why there is no clearsign of stripes in electron-doped materials and could possiblybe related to the much smaller pseudogap energy scale, asdiscussed in Refs. 8 and 9.

Contrary to the phase-separation energy, the AF and SCorder parameters M and D plotted in Fig. 4 only display arather weak cluster-size dependence. This shows that alreadya small 2$2 cluster describes the static ground-state quan-

FIG. 2. !Color online" Chemical potential ! as function of holedoping x. Results for Lc=4 !2$2", Lc=8 !4$2", and Lc=10 clus-ters. The horizontal dashed lines mark the critical !c, and the ver-tical dotted lines mark the boundaries x1 and x2 of the phase sepa-ration region in between.

TABLE I. Discontinuities "x, "M, and "D across the PS regionfor hole and electron doping.

h-Doping "x "M "D

Lc=4 0.115 0.717 0.055Lc=8 0.094 0.699 0.043Lc=10 0.056 0.568 0.032

e-Doping "x "M "D

Lc=4 0.079 0.476 0.016Lc=8 0.020 0.316 0.004Lc=10 0.000 0.000 0.000

FIG. 3. !Color online" Same as Fig. 2 but for electron doping.Note that there is no phase separation for Lc=10; the dotted linemarks the quantum critical point.

FIG. 4. !Color online" Magnetization M and d-wave orderparameter D as functions of hole and electron doping for Lc=4, 8,and 10.

AICHHORN et al. PHYSICAL REVIEW B 74, 235117 !2006"

235117-4

... and what could be achieved(© W. Hanke)

:-(

the main idea of a cluster approach

= +


1 Motivation


= +


(t) + H1


= H0

(t0) +H1



c


• solve the cluster problem exactly!• use the solution to reconstruct

the solution for the full problem(this is approximate !)!

• find a clever way how to do thisstep (“embedding problem”)!

• “divide and conquer strategy”

:-)

cluster-perturbation theory (CPT)


1 Motivation


= +


(t) + H1


= H0

(t0) +H1



c


original system!(L sites)

reference system!(L/Lc clusters with Lc sites each)

hopping matrix t hopping matrix t’ inter-cluster hopping V

t = t0 +V treat this term !perturbatively !

“free” system and Green’s functions

• free system:





H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)






H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)






H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)


G0(!) =1

! + µ� t0 �V=

1

! + µ� t0+

1

! + µ� t0V

1

! + µ� t0+ · · ·

G0(!) = G00(!) +G0

0(!)V G00(!) + · · ·

• reference system:

• free Green’s function • Green’s function of the ref. system:

we have:!

or:!

sum all orders: !

solve: !





H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)






H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)


the “free” CPT equation !!

H0 =X

R1R2�

tR1R2c†R1�

cR2�

<latexit sha1_base64="vZ1CLf0aVaukdwu6GzhoVtBq3G0=">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</latexit><latexit sha1_base64="vZ1CLf0aVaukdwu6GzhoVtBq3G0=">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</latexit><latexit sha1_base64="vZ1CLf0aVaukdwu6GzhoVtBq3G0=">AAACiHicjVFdS8MwFE3r14xfVR99CQ5BEEY7FPVBGPqyRxWnwlpLmmY1LGlLkgqj9Lf4n3zz35huFXTzwQshh3POzb25N8o5U9p1Py17aXllda21Djc2t7Z3nN29R5UVktAByXgmnyOsKGcpHWimOX3OJcUi4vQpGt/U+tMblYpl6YOe5DQQOEnZiBGsDRU67/3QRfAK+qoQYVn6UcZjNRHmQvdV6M0TXV+xROAKQf0fdwXJix/jJKHyD3vzFiSL2ncdGDptt+NOAy0CrwFt0MRt6Hz4cUYKQVNNOFZq6Lm5DkosNSOcVtAvFM0xGeOEDg1MsaAqKKeDrNCRYWI0yqQ5qUZT9mdGiYWqmzROgfWrmtdq8i9tWOjRRVCyNC80Tcms0KjgSGeo3gqKmaRE84kBmEhmekXkFUtMtNldPQRv/suL4LHb8dyOd3fa7nWbcbTAATgEx8AD56AH+uAWDACxVqwT69Q6s6Ht2uf25cxqW03OPvgV9vUXEPrFbQ==</latexit><latexit sha1_base64="vZ1CLf0aVaukdwu6GzhoVtBq3G0=">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</latexit>

H 00 =

X

R1R2�

t0R1R2c†R1�

cR2�

<latexit sha1_base64="NKlPgrc9qwgttDOaG7qtuOhVkLM=">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</latexit><latexit sha1_base64="NKlPgrc9qwgttDOaG7qtuOhVkLM=">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</latexit><latexit sha1_base64="NKlPgrc9qwgttDOaG7qtuOhVkLM=">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</latexit><latexit sha1_base64="NKlPgrc9qwgttDOaG7qtuOhVkLM=">AAACinicjVFdS8MwFE3rx2adOvXRl+CQ+eJohzhFBEEffJzi3GCdJU2zLixpS5IKo/TH+Jd889+YbhV088ELIYdzzs29uddPGJXKtj8Nc219Y7NS3bK2azu7e/X9gxcZpwKTHo5ZLAY+koTRiPQUVYwMEkEQ9xnp+9O7Qu+/ESFpHD2rWUJGHIURHVOMlKa8+vtD07OhdWO5MuVelrl+zAI54/qCT7nnLBNtV9KQoxxaqvkfe27hVzdAYUjEH/byMQuvat+FLK/esFv2POAqcErQAGV0vfqHG8Q45SRSmCEph46dqFGGhKKYkdxyU0kShKcoJEMNI8SJHGXzUebwRDMBHMdCn0jBOfszI0NcFk1qJ0dqIpe1gvxLG6ZqfDnKaJSkikR4UWicMqhiWOwFBlQQrNhMA4QF1b1CPEECYaW3VwzBWf7yKnhptxy75TyeN27b5Tiq4Agcg1PggA64BQ+gC3oAGxXjzLgwOmbNbJtX5vXCahplziH4Feb9FxBCxc8=</latexit>

CPT for interacting systems

• “free” CPT equation:

• CPT equation:!





H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)






H1

=

U

2

X

i�

ni�ni�� (9)



= H0

(t0)+H1


0

(t) + H1


G(!) = G0(!) +G0

(!)V G(!) (10)


c





Gij�(!) =

Z 1

�1dz

Aij�(z)

! � z, (11)


Aij�(z) =

Z 1



Aij�(t) =1

2⇡h[ci�(t), c†j�(0)]+i (13)



(exact)

(approximate)

Gros, Valenti (1993), Senechal et al. (2000)

CPT:!• provides interacting G for (almost) arbitrarily large systems (large L)!• (in principle) controlled by 1/LC (with LC: number of cluster sites)!• with LC=1, this is the “Hubbard-I approximation” !

Hubbard (1963)

CPT: alternative derivation

• Dyson equation of the reference systems

• Dyson equation of the original system:

G0(!) = G00(!) +G0

0(!)⌃0(!)G0(!)

<latexit sha1_base64="sOeInW5NVAmPj28d0pip6rl8u38=">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</latexit><latexit sha1_base64="sOeInW5NVAmPj28d0pip6rl8u38=">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</latexit><latexit sha1_base64="sOeInW5NVAmPj28d0pip6rl8u38=">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</latexit><latexit sha1_base64="sOeInW5NVAmPj28d0pip6rl8u38=">AAACeXicfZHLSgMxFIYz462Ot6pLXURLtSqUmSroRii40GVFe4FOKZk0nYYmkyHJCGXoO/hs7nwRN25ML4te1AMhP/+Xwzk5J4gZVdp1Py17ZXVtfSOz6Wxt7+zuZfcPakokEpMqFkzIRoAUYTQiVU01I41YEsQDRupB/2HE629EKiqiVz2ISYujMKJdipE2Vjv7nvqBYB014OaCj8Pzgi84CdEFdO6hswDb7gy++h/PMf+FhhwN/8KzZdvZnFt0xwGXhTcVOTCNSjv74XcETjiJNGZIqabnxrqVIqkpZmTo+IkiMcJ9FJKmkRHiRLXS8eSGMG+cDuwKaU6k4didzUgRV6MmzUuOdE8tspH5G2smunvXSmkUJ5pEeFKomzCoBRytAXaoJFizgREIS2p6hbiHJMLaLMsxQ/AWv7wsaqWi5xa955tcuTQdRwYcgVNQAB64BWXwBCqgCjD4so6tvHVmfdsndsG+nDy1rWnOIZgL+/oHI1q79Q==</latexit>

G(!) = G0(!) +G0(!)⌃(!)G(!)<latexit sha1_base64="+nhJ/XIaNljMsrJLBIJ7isKrQy0=">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</latexit><latexit sha1_base64="+nhJ/XIaNljMsrJLBIJ7isKrQy0=">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</latexit><latexit sha1_base64="+nhJ/XIaNljMsrJLBIJ7isKrQy0=">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</latexit><latexit sha1_base64="+nhJ/XIaNljMsrJLBIJ7isKrQy0=">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</latexit>

• assume that !⌃0(!) = ⌃(!)

<latexit sha1_base64="9RhO74P1GK2TLSOm5woeHDkrUKg=">AAACKHicbVDLSgMxFM34rOOr6tJNsIh1U2aKoBux4MZlRfuATimZNG1D8xiSjFCGfo4bf8WNiCLd+iVm2gG17YWQwznncu89YcSoNp43cVZW19Y3NnNb7vbO7t5+/uCwrmWsMKlhyaRqhkgTRgWpGWoYaUaKIB4y0giHt6neeCJKUykezSgibY76gvYoRsZSnfxNEoSSdfWI2w8GD7TP0fisGEhO+ugcutfuMsOv3skXvJI3LbgI/AwUQFbVTv496EoccyIMZkjrlu9Fpp0gZShmZOwGsSYRwkPUJy0LBeJEt5PpoWN4apku7EllnzBwyv7tSBDX6abWyZEZ6HktJZdprdj0rtoJFVFsiMCzQb2YQSNhmhrsUkWwYSMLEFbU7grxACmEjc02DcGfP3kR1Msl3yv59xeFSjmLIweOwQkoAh9cggq4A1VQAxg8g1fwAT6dF+fN+XImM+uKk/UcgX/lfP8A+ESlLw==</latexit><latexit sha1_base64="9RhO74P1GK2TLSOm5woeHDkrUKg=">AAACKHicbVDLSgMxFM34rOOr6tJNsIh1U2aKoBux4MZlRfuATimZNG1D8xiSjFCGfo4bf8WNiCLd+iVm2gG17YWQwznncu89YcSoNp43cVZW19Y3NnNb7vbO7t5+/uCwrmWsMKlhyaRqhkgTRgWpGWoYaUaKIB4y0giHt6neeCJKUykezSgibY76gvYoRsZSnfxNEoSSdfWI2w8GD7TP0fisGEhO+ugcutfuMsOv3skXvJI3LbgI/AwUQFbVTv496EoccyIMZkjrlu9Fpp0gZShmZOwGsSYRwkPUJy0LBeJEt5PpoWN4apku7EllnzBwyv7tSBDX6abWyZEZ6HktJZdprdj0rtoJFVFsiMCzQb2YQSNhmhrsUkWwYSMLEFbU7grxACmEjc02DcGfP3kR1Msl3yv59xeFSjmLIweOwQkoAh9cggq4A1VQAxg8g1fwAT6dF+fN+XImM+uKk/UcgX/lfP8A+ESlLw==</latexit><latexit sha1_base64="9RhO74P1GK2TLSOm5woeHDkrUKg=">AAACKHicbVDLSgMxFM34rOOr6tJNsIh1U2aKoBux4MZlRfuATimZNG1D8xiSjFCGfo4bf8WNiCLd+iVm2gG17YWQwznncu89YcSoNp43cVZW19Y3NnNb7vbO7t5+/uCwrmWsMKlhyaRqhkgTRgWpGWoYaUaKIB4y0giHt6neeCJKUykezSgibY76gvYoRsZSnfxNEoSSdfWI2w8GD7TP0fisGEhO+ugcutfuMsOv3skXvJI3LbgI/AwUQFbVTv496EoccyIMZkjrlu9Fpp0gZShmZOwGsSYRwkPUJy0LBeJEt5PpoWN4apku7EllnzBwyv7tSBDX6abWyZEZ6HktJZdprdj0rtoJFVFsiMCzQb2YQSNhmhrsUkWwYSMLEFbU7grxACmEjc02DcGfP3kR1Msl3yv59xeFSjmLIweOwQkoAh9cggq4A1VQAxg8g1fwAT6dF+fN+XImM+uKk/UcgX/lfP8A+ESlLw==</latexit><latexit sha1_base64="9RhO74P1GK2TLSOm5woeHDkrUKg=">AAACKHicbVDLSgMxFM34rOOr6tJNsIh1U2aKoBux4MZlRfuATimZNG1D8xiSjFCGfo4bf8WNiCLd+iVm2gG17YWQwznncu89YcSoNp43cVZW19Y3NnNb7vbO7t5+/uCwrmWsMKlhyaRqhkgTRgWpGWoYaUaKIB4y0giHt6neeCJKUykezSgibY76gvYoRsZSnfxNEoSSdfWI2w8GD7TP0fisGEhO+ugcutfuMsOv3skXvJI3LbgI/AwUQFbVTv496EoccyIMZkjrlu9Fpp0gZShmZOwGsSYRwkPUJy0LBeJEt5PpoWN4apku7EllnzBwyv7tSBDX6abWyZEZ6HktJZdprdj0rtoJFVFsiMCzQb2YQSNhmhrsUkWwYSMLEFbU7grxACmEjc02DcGfP3kR1Msl3yv59xeFSjmLIweOwQkoAh9cggq4A1VQAxg8g1fwAT6dF+fN+XImM+uKk/UcgX/lfP8A+ESlLw==</latexit>

(approximate)

• this yields !G0

0(!)�1 �G0(!)�1 = G0(!)

�1 �G(!)�1<latexit sha1_base64="5EBJ8ni9htXEimG97ueiNNz00AE=">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</latexit><latexit sha1_base64="5EBJ8ni9htXEimG97ueiNNz00AE=">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</latexit><latexit sha1_base64="5EBJ8ni9htXEimG97ueiNNz00AE=">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</latexit><latexit sha1_base64="5EBJ8ni9htXEimG97ueiNNz00AE=">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</latexit>

G(!)�1 = G0(!)�1 �G00(!)

�1 +G0(!)�1 = G0(!)�1 � (! � t0) + (! � t)

<latexit sha1_base64="isf9S39sQnGCDjP1HFxwWJtjEn0=">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</latexit><latexit sha1_base64="isf9S39sQnGCDjP1HFxwWJtjEn0=">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</latexit><latexit sha1_base64="isf9S39sQnGCDjP1HFxwWJtjEn0=">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</latexit><latexit sha1_base64="isf9S39sQnGCDjP1HFxwWJtjEn0=">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</latexit>

G(!)�1 = G0(!)�1 � V<latexit sha1_base64="lIwfrL5mlYcTRytTHGXyYCwiLms=">AAACO3icbVDLSgMxFM34rONr1KWbYBHromWmCLoRCi50WcU+oK0lk6ZtaDIZkoxQhvkvN/6EOzduXCji1r2Zdha29UDI4Zx7ufceP2RUadd9tZaWV1bX1nMb9ubW9s6us7dfVyKSmNSwYEI2faQIowGpaaoZaYaSIO4z0vBHV6nfeCRSURHc63FIOhwNAtqnGGkjdZ27uO0L1lNjbj54nRTagpMBOn2Ii15iX9pz9smsX5z164lt0HXybsmdAC4SLyN5kKHadV7aPYEjTgKNGVKq5bmh7sRIaooZSex2pEiI8AgNSMvQAHGiOvHk9gQeG6UH+0KaF2g4Uf92xIirdD1TyZEeqnkvFf/zWpHuX3RiGoSRJgGeDupHDGoB0yBhj0qCNRsbgrCkZleIh0girE3caQje/MmLpF4ueW7Juz3LV8pZHDlwCI5AAXjgHFTADaiCGsDgCbyBD/BpPVvv1pf1PS1dsrKeAzAD6+cX58yraQ==</latexit><latexit sha1_base64="lIwfrL5mlYcTRytTHGXyYCwiLms=">AAACO3icbVDLSgMxFM34rONr1KWbYBHromWmCLoRCi50WcU+oK0lk6ZtaDIZkoxQhvkvN/6EOzduXCji1r2Zdha29UDI4Zx7ufceP2RUadd9tZaWV1bX1nMb9ubW9s6us7dfVyKSmNSwYEI2faQIowGpaaoZaYaSIO4z0vBHV6nfeCRSURHc63FIOhwNAtqnGGkjdZ27uO0L1lNjbj54nRTagpMBOn2Ii15iX9pz9smsX5z164lt0HXybsmdAC4SLyN5kKHadV7aPYEjTgKNGVKq5bmh7sRIaooZSex2pEiI8AgNSMvQAHGiOvHk9gQeG6UH+0KaF2g4Uf92xIirdD1TyZEeqnkvFf/zWpHuX3RiGoSRJgGeDupHDGoB0yBhj0qCNRsbgrCkZleIh0girE3caQje/MmLpF4ueW7Juz3LV8pZHDlwCI5AAXjgHFTADaiCGsDgCbyBD/BpPVvv1pf1PS1dsrKeAzAD6+cX58yraQ==</latexit><latexit sha1_base64="lIwfrL5mlYcTRytTHGXyYCwiLms=">AAACO3icbVDLSgMxFM34rONr1KWbYBHromWmCLoRCi50WcU+oK0lk6ZtaDIZkoxQhvkvN/6EOzduXCji1r2Zdha29UDI4Zx7ufceP2RUadd9tZaWV1bX1nMb9ubW9s6us7dfVyKSmNSwYEI2faQIowGpaaoZaYaSIO4z0vBHV6nfeCRSURHc63FIOhwNAtqnGGkjdZ27uO0L1lNjbj54nRTagpMBOn2Ii15iX9pz9smsX5z164lt0HXybsmdAC4SLyN5kKHadV7aPYEjTgKNGVKq5bmh7sRIaooZSex2pEiI8AgNSMvQAHGiOvHk9gQeG6UH+0KaF2g4Uf92xIirdD1TyZEeqnkvFf/zWpHuX3RiGoSRJgGeDupHDGoB0yBhj0qCNRsbgrCkZleIh0girE3caQje/MmLpF4ueW7Juz3LV8pZHDlwCI5AAXjgHFTADaiCGsDgCbyBD/BpPVvv1pf1PS1dsrKeAzAD6+cX58yraQ==</latexit><latexit sha1_base64="lIwfrL5mlYcTRytTHGXyYCwiLms=">AAACO3icbVDLSgMxFM34rONr1KWbYBHromWmCLoRCi50WcU+oK0lk6ZtaDIZkoxQhvkvN/6EOzduXCji1r2Zdha29UDI4Zx7ufceP2RUadd9tZaWV1bX1nMb9ubW9s6us7dfVyKSmNSwYEI2faQIowGpaaoZaYaSIO4z0vBHV6nfeCRSURHc63FIOhwNAtqnGGkjdZ27uO0L1lNjbj54nRTagpMBOn2Ii15iX9pz9smsX5z164lt0HXybsmdAC4SLyN5kKHadV7aPYEjTgKNGVKq5bmh7sRIaooZSex2pEiI8AgNSMvQAHGiOvHk9gQeG6UH+0KaF2g4Uf92xIirdD1TyZEeqnkvFf/zWpHuX3RiGoSRJgGeDupHDGoB0yBhj0qCNRsbgrCkZleIh0girE3caQje/MmLpF4ueW7Juz3LV8pZHDlwCI5AAXjgHFTADaiCGsDgCbyBD/BpPVvv1pf1PS1dsrKeAzAD6+cX58yraQ==</latexit>

G(!) = G0(!) +G0(!)V G(!)<latexit sha1_base64="dBK4o2/5/hzFCSBCdkIlGoqVrZc=">AAACZHicfVFdS8MwFE3rx2ad2jl8EiQ4xIkw2iHoizDwQR8nuA/YxkjTdAtLm5Kkwij7k7756Iu/w2wr6jrxQsjhnHO5NydezKhUjvNumFvbO7uF4p61Xzo4PLLLxx3JE4FJG3PGRc9DkjAakbaiipFeLAgKPUa63vRhoXdfiZCURy9qFpNhiMYRDShGSlMjO00HHme+nIX6go/z2oCHZIyuoHUPrZx2+SNe/yeuKZ153vltHNlVp+4sC24CNwNVkFVrZL8NfI6TkEQKMyRl33ViNUyRUBQzMrcGiSQxwlM0Jn0NIxQSOUyXIc3hhWZ8GHChT6Tgkv3dkaJQLnbUzhCpicxrC/IvrZ+o4G6Y0ihOFInwalCQMKg4XCQOfSoIVmymAcKC6l0hniCBsNL/YukQ3PyTN0GnUXeduvt8U202sjiK4BScgxpwwS1ogifQAm2AwYdRMGyjbHyaJbNinqysppH1VMBamWdfz4C0ww==</latexit><latexit sha1_base64="dBK4o2/5/hzFCSBCdkIlGoqVrZc=">AAACZHicfVFdS8MwFE3rx2ad2jl8EiQ4xIkw2iHoizDwQR8nuA/YxkjTdAtLm5Kkwij7k7756Iu/w2wr6jrxQsjhnHO5NydezKhUjvNumFvbO7uF4p61Xzo4PLLLxx3JE4FJG3PGRc9DkjAakbaiipFeLAgKPUa63vRhoXdfiZCURy9qFpNhiMYRDShGSlMjO00HHme+nIX6go/z2oCHZIyuoHUPrZx2+SNe/yeuKZ153vltHNlVp+4sC24CNwNVkFVrZL8NfI6TkEQKMyRl33ViNUyRUBQzMrcGiSQxwlM0Jn0NIxQSOUyXIc3hhWZ8GHChT6Tgkv3dkaJQLnbUzhCpicxrC/IvrZ+o4G6Y0ihOFInwalCQMKg4XCQOfSoIVmymAcKC6l0hniCBsNL/YukQ3PyTN0GnUXeduvt8U202sjiK4BScgxpwwS1ogifQAm2AwYdRMGyjbHyaJbNinqysppH1VMBamWdfz4C0ww==</latexit><latexit sha1_base64="dBK4o2/5/hzFCSBCdkIlGoqVrZc=">AAACZHicfVFdS8MwFE3rx2ad2jl8EiQ4xIkw2iHoizDwQR8nuA/YxkjTdAtLm5Kkwij7k7756Iu/w2wr6jrxQsjhnHO5NydezKhUjvNumFvbO7uF4p61Xzo4PLLLxx3JE4FJG3PGRc9DkjAakbaiipFeLAgKPUa63vRhoXdfiZCURy9qFpNhiMYRDShGSlMjO00HHme+nIX6go/z2oCHZIyuoHUPrZx2+SNe/yeuKZ153vltHNlVp+4sC24CNwNVkFVrZL8NfI6TkEQKMyRl33ViNUyRUBQzMrcGiSQxwlM0Jn0NIxQSOUyXIc3hhWZ8GHChT6Tgkv3dkaJQLnbUzhCpicxrC/IvrZ+o4G6Y0ihOFInwalCQMKg4XCQOfSoIVmymAcKC6l0hniCBsNL/YukQ3PyTN0GnUXeduvt8U202sjiK4BScgxpwwS1ogifQAm2AwYdRMGyjbHyaJbNinqysppH1VMBamWdfz4C0ww==</latexit><latexit sha1_base64="dBK4o2/5/hzFCSBCdkIlGoqVrZc=">AAACZHicfVFdS8MwFE3rx2ad2jl8EiQ4xIkw2iHoizDwQR8nuA/YxkjTdAtLm5Kkwij7k7756Iu/w2wr6jrxQsjhnHO5NydezKhUjvNumFvbO7uF4p61Xzo4PLLLxx3JE4FJG3PGRc9DkjAakbaiipFeLAgKPUa63vRhoXdfiZCURy9qFpNhiMYRDShGSlMjO00HHme+nIX6go/z2oCHZIyuoHUPrZx2+SNe/yeuKZ153vltHNlVp+4sC24CNwNVkFVrZL8NfI6TkEQKMyRl33ViNUyRUBQzMrcGiSQxwlM0Jn0NIxQSOUyXIc3hhWZ8GHChT6Tgkv3dkaJQLnbUzhCpicxrC/IvrZ+o4G6Y0ihOFInwalCQMKg4XCQOfSoIVmymAcKC6l0hniCBsNL/YukQ3PyTN0GnUXeduvt8U202sjiK4BScgxpwwS1ogifQAm2AwYdRMGyjbHyaJbNinqysppH1VMBamWdfz4C0ww==</latexit>

(CPT equation)

CPT in practice

• compute Green’s function of the reference system, e.g., by exact diag.:

• CPT equation:!





H1

=

U

2

X

i�

ni�ni�� (9)



= H0

(t0)+H1


0

(t) + H1


G(!) = G0(!) +G0

(!)V G(!) (10)


c





Gij�(!) =

Z 1

�1dz

Aij�(z)

! � z, (11)


Aij�(z) =

Z 1



Aij�(t) =1

2⇡h[ci�(t), c†j�(0)]+i (13)







H0

=

X

ij�


0

(t) . (2)


H 00

=

X

ij�


0

(t0) , (3)



0

and Lc


c


t = t0 + V (4)


G0

(!) =1

! + µ� t. (5)


G00

(!) =1

! + µ� t0, (6)


G0

(!) = G00

(!) +G00

(!)V G0

(!) (7)

which is solved by

G0

(!) =1

G00

(!)�1 � V. (8)


with

• solve by matrix inversion for any frequency:!

G(!) =1

G0(!)�1 � V

� = 1/T

• partition function

Z 0 =X

m

e��E0m

G0ij�(!) =

1

Z 0

X

mn

(e��E0m + e��E0

n)hm0|ci�|n0ihn0|c†j�|m0i! � (E0

n � E0m)

(H 0 � µN)|n0i = E0n|n0i

QMC vs. CPT

assembling clusters

0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2


0

42

−4−6−8

ΓMXΓk

8

−2

6E

−8−6−4

2

−20

68

4

0

42

−4−6−8

ΓMXΓk

8

−2

6ω

−8−6−4

2

−20

68

4

Γ X M Γ

ω

ω

CPT

QMC

V−CPT


2D Hubbard model, n=1, U=8

L=8x8 sites, T=0.1, QMC L=O(104) sites, Lc=10, T=0, CPT

strange cluster?

strange cluster? but it works!

strange cluster? but it works!

PERIODIZATION


7 Periodization schemes

Note that there is an obvious problem, which is actually shared by any cluster approach that isformulated in real space. Namely, as the reference system is given by a set of decoupled clus-ters, the approximate self-energy does not preserve the translational symmetries of the originallattice. The CPT Green’s function G

R,R0(!) = G e

R

eR

0,rr0(!) is merely invariant under superlat-tice translations e

R ! eR +� e

R. (Note that we use a notation with the spin-projection index �

suppressed). This means G eR

eR

0,rr0(!) = G eR+� e

R, eR0+� eR,r,r0(!). Hence, Fourier transformation

V , see Eq. (12), diagonalizes G with respect to the superlattice indices:

Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)

Here, the V -Fourier transform t(

ek) of the hopping matrix t e

R

eR

0,rr0 is an Lc ⇥ Lc matrix withelements t

rr

0(

ek) for each wave vector e

k in the reciprocal supercell. The self-energy is “cluster-local”, i.e., it is diagonal in the superlattice indices e

R, eR

0, and thus, after V -Fourier transfor-mation, is independent of e

k. The “cluster-local” elements of G with eR =

eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)

From the fully local elements G(loc)rr

(!) we then get the local interacting density of states(LDOS) at a site R = (

eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)

One would expect that the LDOS exhibits the same (translational) symmetries as the Hamil-tonian of the original system and of the original lattice: A

R

(!) = A(!). Within the CPT,however, there is an artificial dependence of the LDOS on r.

A modified cluster-perturbation theory, constructed in the same way as the standard CPT butconsidering periodic boundary conditions for each of the individual clusters, has been suggestedby Zacher et al. [28]. It has been recognized [29], however, that this gives less convincing resultsin practice. The modified CPT makes the Green’s function invariant under translations withineach cluster. However, since U 6= V W , there is also no improvement on the formal level.

One could also try to transform the original model and the reference system as well to reciprocalspace. This straightforward idea means to express H and H 0 in terms of annihilators c

k

ratherthan c

R

and analogously for the creators and to employ the CPT decoupling thereafter. Clearly,H 0 will exhibit the full translational symmetry. However, the idea will not work for models withlocal interactions, such as the Hubbard model. A local interaction in real space transforms into




R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





















reciprocal space





translation symmetries

CPT Green’s function

invariant under superlattice translations only:




R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R


(partial) diagonalization by superlattice Fourier transformation V

cluster-local elements: local density of states:

6= A(!)<latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit><latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit><latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit><latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit>




R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R





R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R


translation symmetries

CPT Green’s function

invariant under superlattice translations only:




R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R


(partial) diagonalization by superlattice Fourier transformation V




















reciprocal space





cluster-local elements: local density of states:

6= A(!)<latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit><latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit><latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit><latexit sha1_base64="Av/f76rXOcqlG3CSgDw7opgVfC8=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BItQL2W3CHqsePFYwX5AdynZdLYNzSZrki2Upb/DiwdFvPpjvPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2lqmikKLSi5VNyQaOBPQMsxw6CYKSBxy6ITju7nfmYDSTIpHM00giMlQsIhRYqwU+ALwbdWXMQzJZb9ccWvuAnideDmpoBzNfvnLH0iaxiAM5UTrnucmJsiIMoxymJX8VENC6JgMoWepIDHoIFscPcMXVhngSCpbwuCF+nsiI7HW0zi0nTExI73qzcX/vF5qopsgYyJJDQi6XBSlHBuJ5wngAVNADZ9aQqhi9lZMR0QRamxOJRuCt/ryOmnXa55b8x6uKo16HkcRnaFzVEUeukYNdI+aqIUoekLP6BW9ORPnxXl3PpatBSefOUV/4Hz+AHrukTI=</latexit>

how to restore the translation symmetry?

(1) use periodic boundary conditions on each cluster Zacher et al. (2000)

AR(!) = A(!)<latexit sha1_base64="ujb2+samf92pcslHC5cxW9qlJ9M=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6VQNyUpgm6EFjcuq9hWaEKYTCbt0EkmzEyEEvoFbvwVNy4UcevanX/jpI2grQeGOZxzL/fe4yeMSmVZX0ZpZXVtfaO8Wdna3tndM/cPepKnApMu5oyLOx9JwmhMuooqRu4SQVDkM9L3x5e5378nQlIe36pJQtwIDWMaUoyUljyz1vYyx+cskJNIf/BmWnd4RIboBF7A9g/3zKrVsGaAy8QuSBUU6HjmpxNwnEYkVpghKQe2lSg3Q0JRzMi04qSSJAiP0ZAMNI1RRKSbzc6ZwppWAhhyoV+s4Ez93ZGhSObr6soIqZFc9HLxP2+QqvDczWicpIrEeD4oTBlUHObZwIAKghWbaIKwoHpXiEdIIKx0ghUdgr148jLpNRu21bCvT6utZhFHGRyBY1AHNjgDLXAFOqALMHgAT+AFvBqPxrPxZrzPS0tG0XMI/sD4+AYtx5rU</latexit><latexit sha1_base64="ujb2+samf92pcslHC5cxW9qlJ9M=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6VQNyUpgm6EFjcuq9hWaEKYTCbt0EkmzEyEEvoFbvwVNy4UcevanX/jpI2grQeGOZxzL/fe4yeMSmVZX0ZpZXVtfaO8Wdna3tndM/cPepKnApMu5oyLOx9JwmhMuooqRu4SQVDkM9L3x5e5378nQlIe36pJQtwIDWMaUoyUljyz1vYyx+cskJNIf/BmWnd4RIboBF7A9g/3zKrVsGaAy8QuSBUU6HjmpxNwnEYkVpghKQe2lSg3Q0JRzMi04qSSJAiP0ZAMNI1RRKSbzc6ZwppWAhhyoV+s4Ez93ZGhSObr6soIqZFc9HLxP2+QqvDczWicpIrEeD4oTBlUHObZwIAKghWbaIKwoHpXiEdIIKx0ghUdgr148jLpNRu21bCvT6utZhFHGRyBY1AHNjgDLXAFOqALMHgAT+AFvBqPxrPxZrzPS0tG0XMI/sD4+AYtx5rU</latexit><latexit sha1_base64="ujb2+samf92pcslHC5cxW9qlJ9M=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6VQNyUpgm6EFjcuq9hWaEKYTCbt0EkmzEyEEvoFbvwVNy4UcevanX/jpI2grQeGOZxzL/fe4yeMSmVZX0ZpZXVtfaO8Wdna3tndM/cPepKnApMu5oyLOx9JwmhMuooqRu4SQVDkM9L3x5e5378nQlIe36pJQtwIDWMaUoyUljyz1vYyx+cskJNIf/BmWnd4RIboBF7A9g/3zKrVsGaAy8QuSBUU6HjmpxNwnEYkVpghKQe2lSg3Q0JRzMi04qSSJAiP0ZAMNI1RRKSbzc6ZwppWAhhyoV+s4Ez93ZGhSObr6soIqZFc9HLxP2+QqvDczWicpIrEeD4oTBlUHObZwIAKghWbaIKwoHpXiEdIIKx0ghUdgr148jLpNRu21bCvT6utZhFHGRyBY1AHNjgDLXAFOqALMHgAT+AFvBqPxrPxZrzPS0tG0XMI/sD4+AYtx5rU</latexit><latexit sha1_base64="ujb2+samf92pcslHC5cxW9qlJ9M=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6VQNyUpgm6EFjcuq9hWaEKYTCbt0EkmzEyEEvoFbvwVNy4UcevanX/jpI2grQeGOZxzL/fe4yeMSmVZX0ZpZXVtfaO8Wdna3tndM/cPepKnApMu5oyLOx9JwmhMuooqRu4SQVDkM9L3x5e5378nQlIe36pJQtwIDWMaUoyUljyz1vYyx+cskJNIf/BmWnd4RIboBF7A9g/3zKrVsGaAy8QuSBUU6HjmpxNwnEYkVpghKQe2lSg3Q0JRzMi04qSSJAiP0ZAMNI1RRKSbzc6ZwppWAhhyoV+s4Ez93ZGhSObr6soIqZFc9HLxP2+QqvDczWicpIrEeD4oTBlUHObZwIAKghWbaIKwoHpXiEdIIKx0ghUdgr148jLpNRu21bCvT6utZhFHGRyBY1AHNjgDLXAFOqALMHgAT+AFvBqPxrPxZrzPS0tG0XMI/sD4+AYtx5rU</latexit>

GeR,eR,r+�r,r0+�r(!) = GeR,eR,r,r0(!)<latexit sha1_base64="3Sgbnlc2Nf5eVDkPI1RAuZVMXSk=">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</latexit><latexit sha1_base64="3Sgbnlc2Nf5eVDkPI1RAuZVMXSk=">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</latexit><latexit sha1_base64="3Sgbnlc2Nf5eVDkPI1RAuZVMXSk=">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</latexit><latexit sha1_base64="3Sgbnlc2Nf5eVDkPI1RAuZVMXSk=">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</latexit>

GR,R(!) = G(!)<latexit sha1_base64="zuVAPJ94DL9FRIGYltpuSw/YXhY=">AAACI3icbVBNS8NAEN3Urxq/qh69LBahgpSkCIogFDzUYxX7AU0Im82mXbrJht2NUEL/ixf/ihcPSvHiwf/ipi2orQ+WffNmhpl5fsKoVJb1aRRWVtfWN4qb5tb2zu5eaf+gLXkqMGlhzrjo+kgSRmPSUlQx0k0EQZHPSMcf3uT5ziMRkvL4QY0S4kaoH9OQYqS05JWuGl6WOT5ngRxF+oP347OFeFxxeET66BSa12bjJ/BKZatqTQGXiT0nZTBH0ytNnIDjNCKxwgxJ2bOtRLkZEopiRsamk0qSIDxEfdLTNEYRkW42vXEMT7QSwJAL/WIFp+rvjgxFMl9ZV0ZIDeRiLhf/y/VSFV66GY2TVJEYzwaFKYOKw9wwGFBBsGIjTRAWVO8K8QAJhJW2NTfBXjx5mbRrVduq2nfn5XptbkcRHIFjUAE2uAB1cAuaoAUweAIv4A28G8/GqzExPmalBWPecwj+wPj6BsnzovE=</latexit><latexit sha1_base64="zuVAPJ94DL9FRIGYltpuSw/YXhY=">AAACI3icbVBNS8NAEN3Urxq/qh69LBahgpSkCIogFDzUYxX7AU0Im82mXbrJht2NUEL/ixf/ihcPSvHiwf/ipi2orQ+WffNmhpl5fsKoVJb1aRRWVtfWN4qb5tb2zu5eaf+gLXkqMGlhzrjo+kgSRmPSUlQx0k0EQZHPSMcf3uT5ziMRkvL4QY0S4kaoH9OQYqS05JWuGl6WOT5ngRxF+oP347OFeFxxeET66BSa12bjJ/BKZatqTQGXiT0nZTBH0ytNnIDjNCKxwgxJ2bOtRLkZEopiRsamk0qSIDxEfdLTNEYRkW42vXEMT7QSwJAL/WIFp+rvjgxFMl9ZV0ZIDeRiLhf/y/VSFV66GY2TVJEYzwaFKYOKw9wwGFBBsGIjTRAWVO8K8QAJhJW2NTfBXjx5mbRrVduq2nfn5XptbkcRHIFjUAE2uAB1cAuaoAUweAIv4A28G8/GqzExPmalBWPecwj+wPj6BsnzovE=</latexit><latexit sha1_base64="zuVAPJ94DL9FRIGYltpuSw/YXhY=">AAACI3icbVBNS8NAEN3Urxq/qh69LBahgpSkCIogFDzUYxX7AU0Im82mXbrJht2NUEL/ixf/ihcPSvHiwf/ipi2orQ+WffNmhpl5fsKoVJb1aRRWVtfWN4qb5tb2zu5eaf+gLXkqMGlhzrjo+kgSRmPSUlQx0k0EQZHPSMcf3uT5ziMRkvL4QY0S4kaoH9OQYqS05JWuGl6WOT5ngRxF+oP347OFeFxxeET66BSa12bjJ/BKZatqTQGXiT0nZTBH0ytNnIDjNCKxwgxJ2bOtRLkZEopiRsamk0qSIDxEfdLTNEYRkW42vXEMT7QSwJAL/WIFp+rvjgxFMl9ZV0ZIDeRiLhf/y/VSFV66GY2TVJEYzwaFKYOKw9wwGFBBsGIjTRAWVO8K8QAJhJW2NTfBXjx5mbRrVduq2nfn5XptbkcRHIFjUAE2uAB1cAuaoAUweAIv4A28G8/GqzExPmalBWPecwj+wPj6BsnzovE=</latexit><latexit sha1_base64="zuVAPJ94DL9FRIGYltpuSw/YXhY=">AAACI3icbVBNS8NAEN3Urxq/qh69LBahgpSkCIogFDzUYxX7AU0Im82mXbrJht2NUEL/ixf/ihcPSvHiwf/ipi2orQ+WffNmhpl5fsKoVJb1aRRWVtfWN4qb5tb2zu5eaf+gLXkqMGlhzrjo+kgSRmPSUlQx0k0EQZHPSMcf3uT5ziMRkvL4QY0S4kaoH9OQYqS05JWuGl6WOT5ngRxF+oP347OFeFxxeET66BSa12bjJ/BKZatqTQGXiT0nZTBH0ytNnIDjNCKxwgxJ2bOtRLkZEopiRsamk0qSIDxEfdLTNEYRkW42vXEMT7QSwJAL/WIFp+rvjgxFMl9ZV0ZIDeRiLhf/y/VSFV66GY2TVJEYzwaFKYOKw9wwGFBBsGIjTRAWVO8K8QAJhJW2NTfBXjx5mbRrVduq2nfn5XptbkcRHIFjUAE2uAB1cAuaoAUweAIv4A28G8/GqzExPmalBWPecwj+wPj6BsnzovE=</latexit>

but:




R,R0(!) = G e

R

eR


R ! eR +� e



eR

0,rr0(!) = G eR+� e



Lc

L

X

eR

fR

0

V †ek, eR

G eR

eR

0,rr0(!)V eR

0,ek0 = Grr

0(

ek,!)�e

k

ek

0 (37)

withG

rr

0(

ek,!) =

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0(38)



R

eR


rr

0(



R, eR



eR

0 are obtainedby

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (39)



eR, r) as

AR

(!) = � 1

⇡ImG

rr

(! + i0+) . (40)


R




k

ratherthan c

R


and:

GR+�R,R0+�R(!) 6= GR,R0(!)<latexit sha1_base64="3J5dLdc2sDobh3qRGaIURaOqq2w=">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</latexit><latexit sha1_base64="3J5dLdc2sDobh3qRGaIURaOqq2w=">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</latexit><latexit sha1_base64="3J5dLdc2sDobh3qRGaIURaOqq2w=">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</latexit><latexit sha1_base64="3J5dLdc2sDobh3qRGaIURaOqq2w=">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</latexit>

U 6= V W<latexit sha1_base64="iV1zj0zZpfmgeteewot9poze310=">AAACHHicbVBNS8MwGE79nPWr6tFLcAieRjsFPQ68eJxgt8FaRppmW1ialCQVRukP8eJf8eJBES8eBP+N6daD23wg5MnzvC953ydKGVXadX+stfWNza3t2o69u7d/cOgcHXeUyCQmPhZMyF6EFGGUE19TzUgvlQQlESPdaHJb+t1HIhUV/EFPUxImaMTpkGKkjTRwLvMgEixW08Rc0C8CTuwFqVMsvruFPXDqbsOdAa4SryJ1UKE9cL6CWOAsIVxjhpTqe26qwxxJTTEjhR1kiqQIT9CI9A3lKCEqzGfLFfDcKDEcCmkO13Cm/u3IUaLK2UxlgvRYLXul+J/Xz/TwJswpTzNNOJ5/NMwY1AKWScGYSoI1mxqCsKRmVojHSCKsTZ5lCN7yyquk02x4bsO7v6q3mlUcNXAKzsAF8MA1aIE70AY+wOAJvIA38G49W6/Wh/U5L12zqp4TsADr+xcdwaHp</latexit><latexit sha1_base64="iV1zj0zZpfmgeteewot9poze310=">AAACHHicbVBNS8MwGE79nPWr6tFLcAieRjsFPQ68eJxgt8FaRppmW1ialCQVRukP8eJf8eJBES8eBP+N6daD23wg5MnzvC953ydKGVXadX+stfWNza3t2o69u7d/cOgcHXeUyCQmPhZMyF6EFGGUE19TzUgvlQQlESPdaHJb+t1HIhUV/EFPUxImaMTpkGKkjTRwLvMgEixW08Rc0C8CTuwFqVMsvruFPXDqbsOdAa4SryJ1UKE9cL6CWOAsIVxjhpTqe26qwxxJTTEjhR1kiqQIT9CI9A3lKCEqzGfLFfDcKDEcCmkO13Cm/u3IUaLK2UxlgvRYLXul+J/Xz/TwJswpTzNNOJ5/NMwY1AKWScGYSoI1mxqCsKRmVojHSCKsTZ5lCN7yyquk02x4bsO7v6q3mlUcNXAKzsAF8MA1aIE70AY+wOAJvIA38G49W6/Wh/U5L12zqp4TsADr+xcdwaHp</latexit><latexit sha1_base64="iV1zj0zZpfmgeteewot9poze310=">AAACHHicbVBNS8MwGE79nPWr6tFLcAieRjsFPQ68eJxgt8FaRppmW1ialCQVRukP8eJf8eJBES8eBP+N6daD23wg5MnzvC953ydKGVXadX+stfWNza3t2o69u7d/cOgcHXeUyCQmPhZMyF6EFGGUE19TzUgvlQQlESPdaHJb+t1HIhUV/EFPUxImaMTpkGKkjTRwLvMgEixW08Rc0C8CTuwFqVMsvruFPXDqbsOdAa4SryJ1UKE9cL6CWOAsIVxjhpTqe26qwxxJTTEjhR1kiqQIT9CI9A3lKCEqzGfLFfDcKDEcCmkO13Cm/u3IUaLK2UxlgvRYLXul+J/Xz/TwJswpTzNNOJ5/NMwY1AKWScGYSoI1mxqCsKRmVojHSCKsTZ5lCN7yyquk02x4bsO7v6q3mlUcNXAKzsAF8MA1aIE70AY+wOAJvIA38G49W6/Wh/U5L12zqp4TsADr+xcdwaHp</latexit><latexit sha1_base64="iV1zj0zZpfmgeteewot9poze310=">AAACHHicbVBNS8MwGE79nPWr6tFLcAieRjsFPQ68eJxgt8FaRppmW1ialCQVRukP8eJf8eJBES8eBP+N6daD23wg5MnzvC953ydKGVXadX+stfWNza3t2o69u7d/cOgcHXeUyCQmPhZMyF6EFGGUE19TzUgvlQQlESPdaHJb+t1HIhUV/EFPUxImaMTpkGKkjTRwLvMgEixW08Rc0C8CTuwFqVMsvruFPXDqbsOdAa4SryJ1UKE9cL6CWOAsIVxjhpTqe26qwxxJTTEjhR1kiqQIT9CI9A3lKCEqzGfLFfDcKDEcCmkO13Cm/u3IUaLK2UxlgvRYLXul+J/Xz/TwJswpTzNNOJ5/NMwY1AKWScGYSoI1mxqCsKRmVojHSCKsTZ5lCN7yyquk02x4bsO7v6q3mlUcNXAKzsAF8MA1aIE70AY+wOAJvIA38G49W6/Wh/U5L12zqp4TsADr+xcdwaHp</latexit>

since:

in practice: results are worse compared to clusters with open b.c.


1 Motivation


= +


(t) + H1


= H0

(t0) +H1



c



(2) take the average


a delocalized one in k-space where the interaction parameters Ukk

0k

00k

00 basically couple any k

point to any other.

With respect to the local spectral density (40), a pragmatic way out would be to average overthe cluster sites:

A(!) ⌘ 1

Lc

X

r

A( eR,r)(!) =1

L

X

R

AR

(!) . (41)

One thus distinguishes between the CPT spectral density on the one hand and the translationallyinvariant (“physical”) spectral density on the other. Generally, a periodization operator b

T canbe defined which, for example, enforces a translationally invariant Green’s function: G(!) 7!bT [G](!). A possible construction of b

T is to start with a U -Fourier transform of the CPTGreen’s function from R-space to k-space. As the Green’s function does not fully respectthe translation symmetries, this transformation does not fully diagonalize the Green’s function,and we get G

kk

0(!) with nonzero elements for k 6= k

0. Periodization is then achieved byreplacing G

kk

0(!) 7! G

kk

(!)�k,k0 ⌘ bT [G]

kk

0(!). This provides us with a translationally

invariant (physical) Green’s function bT [G](!). In real space, this periodization reads:

bT [G]

RR

0=

1

L

X

R

00R

000

�R�R

0,R00�R

000GR

00R

000 . (42)

For R = R

0, in particular, we have

bT [G]

RR

(!) =1

L

X

R

00R

000

�R

00R

000GR

00R

000(!) =

1

L

X

R

GRR

(!) =1

Lc

X

r

Grr

(!) , (43)

consistent with Eq. (41). This periodization is actually a standard procedure and has also beenused to produce the results displayed in Fig. 7. In addition a smoothening of the spectrum hasbeen employed by replacing the positive infinitesimal 0+ with a small but finite value ⌘ > 0 inEq. (19). This is necessary since the Green’s function of the reference system and thus the CPTself-energy consists of a finite number of poles only such that, even after periodization Eq. (41),the LDOS is composed of a finite number of �-peaks only.

A different periodization scheme is given by periodizing the CPT self-energy,

⌃(!) 7! bT [⌃](!) . (44)

Concretely, with ⌃( eR,r),( eR0,r0)(!) = ⌃r,r0

(!)� eR, eR0 , we have

bT [⌃]( eR,r),( eR0,r0)(!) =1

Lc

X

r

00,r000

� eR+r� e

R

0�r

0,r00�r

000⌃r

00,r000(!) . (45)

This produces a self-energy which fully respects the translational symmetries and thus, via Eq.(36), a fully translationally invariant Green’s function. Both periodization schemes share thesame ad hoc character. The periodization of the self-energy appears a bit more artificial, as thenecessary ad hoc approximation is performed at an earlier stage of the theory rather than at thevery end.

“CPT spectral density”“physical spectral density”

(3) periodization



0k

00k


point to any other.


A(!) ⌘ 1

Lc

X

r

A( eR,r)(!) =1

L

X

R

AR

(!) . (41)




kk



kk

0(!) 7! G

kk

(!)�k,k0 ⌘ bT [G]

kk



bT [G]

RR

0=

1

L

X

R

00R

000

�R�R

0,R00�R

000GR

00R

000 . (42)

For R = R


bT [G]

RR

(!) =1

L

X

R

00R

000

�R

00R

000GR

00R

000(!) =

1

L

X

R

GRR

(!) =1

Lc

X

r

Grr

(!) , (43)



⌃(!) 7! bT [⌃](!) . (44)



bT [⌃]( eR,r),( eR0,r0)(!) =1

Lc

X

r

00,r000

� eR+r� e

R

0�r

0,r00�r

000⌃r

00,r000(!) . (45)




0k

00k


point to any other.


A(!) ⌘ 1

Lc

X

r

A( eR,r)(!) =1

L

X

R

AR

(!) . (41)




kk



kk

0(!) 7! G

kk

(!)�k,k0 ⌘ bT [G]

kk



bT [G]

RR

0=

1

L

X

R

00R

000

�R�R

0,R00�R

000GR

00R

000 . (42)

For R = R


bT [G]

RR

(!) =1

L

X

R

00R

000

�R

00R

000GR

00R

000(!) =

1

L

X

R

GRR

(!) =1

Lc

X

r

Grr

(!) , (43)



⌃(!) 7! bT [⌃](!) . (44)



bT [⌃]( eR,r),( eR0,r0)(!) =1

Lc

X

r

00,r000

� eR+r� e

R

0�r

0,r00�r

000⌃r

00,r000(!) . (45)




0k

00k


point to any other.


A(!) ⌘ 1

Lc

X

r

A( eR,r)(!) =1

L

X

R

AR

(!) . (41)




kk



kk

0(!) 7! G

kk

(!)�k,k0 ⌘ bT [G]

kk



bT [G]

RR

0=

1

L

X

R

00R

000

�R�R

0,R00�R

000GR

00R

000 . (42)

For R = R


bT [G]

RR

(!) =1

L

X

R

00R

000

�R

00R

000GR

00R

000(!) =

1

L

X

R

GRR

(!) =1

Lc

X

r

Grr

(!) , (43)



⌃(!) 7! bT [⌃](!) . (44)



bT [⌃]( eR,r),( eR0,r0)(!) =1

Lc

X

r

00,r000

� eR+r� e

R

0�r

0,r00�r

000⌃r

00,r000(!) . (45)


.8 Michael Potthoff


ea1

<latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit><latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit><latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit><latexit sha1_base64="unlvXsAshVP/TCUAuHgjHZq79AM=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KokIdllw47KCfUATwmRy0w6dPJiZKCVk48ZfceNCEbf+gzv/xkmbhbYeGOZwzr3ce4+fciaVZX0bK6tr6xubta369s7u3r55cNiTSSYodGnCEzHwiQTOYugqpjgMUgEk8jn0/cl16ffvQUiWxHdqmoIbkVHMQkaJ0pJnnjgPLADFeAC54yc8kNNIfzkpCs/2zIbVtGbAy8SuSANV6HjmlxMkNIsgVpQTKYe2lSo3J0IxyqGoO5mElNAJGcFQ05hEIN18dkWBz7QS4DAR+sUKz9TfHTmJZLmdroyIGstFrxT/84aZCltuzuI0UxDT+aAw41gluIwEB0wAVXyqCaGC6V0xHRNBqNLB1XUI9uLJy6R30bStpn172Wi3qjhq6BidonNkoyvURjeog7qIokf0jF7Rm/FkvBjvxse8dMWoeo7QHxifP6ormUQ=</latexit> eb1


eb2

<latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit><latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit><latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit><latexit sha1_base64="yArBO+sUsktcCH363309/HT9Eck=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eXBbOY3fbNuN+w5rFXilKSOSnR888sNE5LFwBVhWMqhY6fKy7FQlDCY1dxMQorJBI9gqCnHMUgvn18xs861ElpRIvTjypqrvztyHMtiO10ZYzWWy14h/ucNMxW1vJzyNFPAyWJQlDFLJVYRiRVSAUSxqSaYCKp3tcgYC0yUDq6mQ3CWT14lvWbDsRvO7WW93SrjqKITdIYukIOuUBvdoA7qIoIe0TN6RW/Gk/FivBsfi9KKUfYcoz8wPn8ArTeZRg==</latexit>

ea2

<latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit><latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit><latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit><latexit sha1_base64="R714i3Cx5VN/CiW16BM2JB6qtRY=">AAACBXicbVDLSsNAFJ3UV62vqEtdBIvgqiRFsMuCG5cV7AOaECaTm3boZBJmJkoJ3bjxV9y4UMSt/+DOv3HSZqGtB4Y5nHMv994TpIxKZdvfRmVtfWNzq7pd29nd2z8wD496MskEgS5JWCIGAZbAKIeuoorBIBWA44BBP5hcF37/HoSkCb9T0xS8GI84jSjBSku+eeo+0BAUZSHkbpCwUE5j/eV4NvObvlm3G/Yc1ipxSlJHJTq++eWGCcli4IowLOXQsVPl5VgoShjMam4mIcVkgkcw1JTjGKSXz6+YWedaCa0oEfpxZc3V3x05jmWxna6MsRrLZa8Q//OGmYpaXk55mingZDEoypilEquIxAqpAKLYVBNMBNW7WmSMBSZKB1fTITjLJ6+SXrPh2A3n9rLebpVxVNEJOkMXyEFXqI1uUAd1EUGP6Bm9ojfjyXgx3o2PRWnFKHuO0R8Ynz+rr5lF</latexit>






eR

<latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit>

eG

<latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit><latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit><latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit><latexit sha1_base64="DigxCy+AsQcDcs0ltEIHWdAwFck=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBhS4r2Ac0oUwmN+3QySTMTJQSAm78FTcuFHHrT7jzb5y0WWjrgWEO59zLvff4CaNS2fa3UVlZXVvfqG7WtrZ3dvfM/YOujFNBoENiFou+jyUwyqGjqGLQTwTgyGfQ8ydXhd+7ByFpzO/UNAEvwiNOQ0qw0tLQPHIfaACKsgAy149ZIKeR/rLrPB+adbthz2AtE6ckdVSiPTS/3CAmaQRcEYalHDh2orwMC0UJg7zmphISTCZ4BANNOY5Aetnshtw61UpghbHQjytrpv7uyHAki910ZYTVWC56hfifN0hV2PQyypNUASfzQWHKLBVbRSBWQAUQxaaaYCKo3tUiYywwUTq2mg7BWTx5mXTPG47dcG4v6q1mGUcVHaMTdIYcdIla6Aa1UQcR9Iie0St6M56MF+Pd+JiXVoyy5xD9gfH5A0d/mIY=</latexit>

R = eR + r

<latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit>

�2k

<latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit><latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit><latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit><latexit sha1_base64="j9qPK5La5VF8SYJ3VvDuJRm55FU=">AAACEHicbVDLSsNAFJ34rPUVdekmWEQ3hqQI7bKgC5cV7AOaWCaT23boTBJmJkIJ+QQ3/oobF4q4denOv3HSdqGtB4Y5nHMv994TJIxK5Tjfxsrq2vrGZmmrvL2zu7dvHhy2ZZwKAi0Ss1h0AyyB0QhaiioG3UQA5gGDTjC+KvzOAwhJ4+hOTRLwOR5GdEAJVlrqm2feNTCFvZFMMIHswrFrnOdZfl/1gpiFcsL1l43zvllxbGcKa5m4c1JBczT75pcXxiTlECnCsJQ910mUn2GhKGGQl71Ugh45xkPoaRphDtLPpgfl1qlWQmsQC/0iZU3V3x0Z5rJYTVdyrEZy0SvE/7xeqgZ1P6NRkiqIyGzQIGWWiq0iHSukAohiE00wEVTvapERFpgonWFZh+AunrxM2lXbdWz39rLSqM/jKKFjdILOkYtqqIFuUBO1EEGP6Bm9ojfjyXgx3o2PWemKMe85Qn9gfP4AN7+dTw==</latexit>






reciprocal space<latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit><latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit><latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit><latexit sha1_base64="c3z9t7VtV+Q4g/LcnIwlk3F+NQw=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMiGCXBTcuK9gHtEPJpHfa0GQyJBmxDLPxV9y4UMStn+HOvzFtZ6GtBwKHc+5Nck6YcKaN5307pbX1jc2t8nZlZ3dv/8A9PGprmSoKLSq5VN2QaOAshpZhhkM3UUBEyKETTm5mfucBlGYyvjfTBAJBRjGLGCXGSgP3pC9C+ZgpoCxRkhKOdUIo5AO36tW8OfAq8QtSRQWaA/erP5Q0FRAbyonWPd9LTJARZRjlkFf6qQZ784SMoGdpTAToIJsHyPG5VYY4ksqe2OC5+nsjI0LrqQjtpCBmrJe9mfif10tNVA8yFiepgZguHopSjo3EszbwkNnkhk8tIVQx+1dMx0QRamxnFVuCvxx5lbQva75X8++uqo16UUcZnaIzdIF8dI0a6BY1UQtRlKNn9IrenCfnxXl3PhajJafYOUZ/4Hz+AGkDluE=</latexit>

r

<latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit>

ek

<latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit><latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit><latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit><latexit sha1_base64="3v0P5UMT9AlnBrm8wmqn/L8HbAg=">AAACA3icbVDLSsNAFJ34rPUVdaebYBFclUQEuyy4cVnBPqAJZTK5aYdOZsLMRCkh4MZfceNCEbf+hDv/xmmbhbYeGOZwzr3ce0+YMqq0635bK6tr6xubla3q9s7u3r59cNhRIpME2kQwIXshVsAoh7ammkEvlYCTkEE3HF9P/e49SEUFv9OTFIIEDzmNKcHaSAP72A8Fi9QkMV/uP9AINGUR5OOiGNg1t+7O4CwTryQ1VKI1sL/8SJAsAa4Jw0r1PTfVQY6lpoRBUfUzBSkmYzyEvqEcJ6CCfHZD4ZwZJXJiIc3j2pmpvztynKjpmqYywXqkFr2p+J/Xz3TcCHLK00wDJ/NBccYcLZxpIE5EJRDNJoZgIqnZ1SEjLDHRJraqCcFbPHmZdC7qnlv3bi9rzUYZRwWdoFN0jjx0hZroBrVQGxH0iJ7RK3qznqwX6936mJeuWGXPEfoD6/MHfxmYqg==</latexit>

k = eG + e

k

<latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit><latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit><latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit><latexit sha1_base64="u6DPH4VoSbP+xXbRzi9gKXsmtSk=">AAACMXicdVDLSgMxFM3UV62vUZdugkUQhDIjgt0IBRd2WcE+oC0lk7ltQzOTIckoZZhfcuOfiJsuFHHrT5i2s6itHgg5nHMv997jRZwp7TgTK7e2vrG5ld8u7Ozu7R/Yh0cNJWJJoU4FF7LlEQWchVDXTHNoRRJI4HFoeqPbqd98BKmYCB/0OIJuQAYh6zNKtJF6drXjCe6rcWC+ZJTiG7wodJ6YD5pxH5K7NMUX/5mjNO3ZRafkzIBXiZuRIspQ69mvHV/QOIBQU06UartOpLsJkZpRDmmhEyuICB2RAbQNDUkAqpvMLk7xmVF83BfSvFDjmbrYkZBATdc0lQHRQ7XsTcW/vHas++VuwsIo1hDS+aB+zLEWeBof9pkEqvnYEEIlM7tiOiSSUG1CLpgQ3OWTV0njsuQ6Jff+qlgpZ3Hk0Qk6RefIRdeogqqohuqIomf0ht7Rh/ViTaxP62temrOynmP0C9b3D4bArDU=</latexit>

Fig. 2: Example of a lattice/superlattice in D = 2 dimensions. Real space, left: A systemconsisting of L = 36 sites on a square lattice with periodic boundary conditions. C: unitcell. a1,a2: basis vectors of the lattice. Lines: tiling into clusters with Lc = 4 sites. Orangesites: cluster origins, forming a superlattice structure of L/Lc = 9 clusters. SC: unit cell ofthe superlattice. e

a1, ea2: superlattice basis vectors. Any lattice vector R can be decomposeduniquely into a superlattice vector e

R and a cluster vector r. A1,A2: primitive translationsdefining the periodic boundary conditions and the total system size. Reciprocal space, right:b1, b2 reciprocal basis spanning a reciprocal unit cell RC containing L = 36 k-points. �2k =

(2⇡)/V : volume element. The RC contains Lc = 4 reciprocal supercells (RSC) with L/Lc = 9

wave vectors ek each. Their volume is VRSC = VRC/Lc. Any wave vector k can be decomposed

uniquely into a reciprocal superlattice vector eG and a wave vector e

k.

where eG are the reciprocal superlattice vectors. Note that in a reciprocal unit cell there are L

wave vectors k and Lc reciprocal superlattice vectors eG, and there are L/Lc wave vectors e

k.

Let us consider the L ⇥ L matrix U with elements

UR,k =

1pLeikR , k 2 RC , (11)

and the L/Lc ⇥ L/Lc matrix V with elements

V eR,ek =

1pL/Lc

eiek eR , e

k 2 RSC , (12)

and the Lc ⇥ Lc matrix W with elements

Wr, eG =

1pLc

ei eGr , eG 2 RC . (13)

The unitary matrices U , V and W define discrete Fourier transformations between the respec-tive real and reciprocal spaces. We refer to the different transformations as the lattice Fourier

in k-space:

in real space: bT :<latexit sha1_base64="McN/C1V0wC+ZFgBefHyruFQAwvA=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoHgqePFYoV/QxrLZbNqlm03YnVhK6P/w4kERr/4Xb/4b0zYHbX0w8Hhvhpl5XiyFQdv+tgobm1vbO8Xd0t7+weFR+fikbaJEM95ikYx016OGS6F4CwVK3o01p6Eneccb3839zhPXRkSqidOYuyEdKhEIRjGTHtP+RPh8RJE0Z7elQbliV+0FyDpxclKBHI1B+avvRywJuUImqTE9x47RTalGwSSflfqJ4TFlYzrkvYwqGnLjpourZ+QiU3wSRDorhWSh/p5IaWjMNPSyzpDiyKx6c/E/r5dgcOOmQsUJcsWWi4JEEozIPALiC80ZymlGKNMiu5WwEdWUYRbUPARn9eV10q5VHbvqPFxV6rU8jiKcwTlcggPXUId7aEALGGh4hld4sybWi/VufSxbC1Y+cwp/YH3+AK+Mkeg=</latexit><latexit sha1_base64="McN/C1V0wC+ZFgBefHyruFQAwvA=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoHgqePFYoV/QxrLZbNqlm03YnVhK6P/w4kERr/4Xb/4b0zYHbX0w8Hhvhpl5XiyFQdv+tgobm1vbO8Xd0t7+weFR+fikbaJEM95ikYx016OGS6F4CwVK3o01p6Eneccb3839zhPXRkSqidOYuyEdKhEIRjGTHtP+RPh8RJE0Z7elQbliV+0FyDpxclKBHI1B+avvRywJuUImqTE9x47RTalGwSSflfqJ4TFlYzrkvYwqGnLjpourZ+QiU3wSRDorhWSh/p5IaWjMNPSyzpDiyKx6c/E/r5dgcOOmQsUJcsWWi4JEEozIPALiC80ZymlGKNMiu5WwEdWUYRbUPARn9eV10q5VHbvqPFxV6rU8jiKcwTlcggPXUId7aEALGGh4hld4sybWi/VufSxbC1Y+cwp/YH3+AK+Mkeg=</latexit><latexit sha1_base64="McN/C1V0wC+ZFgBefHyruFQAwvA=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoHgqePFYoV/QxrLZbNqlm03YnVhK6P/w4kERr/4Xb/4b0zYHbX0w8Hhvhpl5XiyFQdv+tgobm1vbO8Xd0t7+weFR+fikbaJEM95ikYx016OGS6F4CwVK3o01p6Eneccb3839zhPXRkSqidOYuyEdKhEIRjGTHtP+RPh8RJE0Z7elQbliV+0FyDpxclKBHI1B+avvRywJuUImqTE9x47RTalGwSSflfqJ4TFlYzrkvYwqGnLjpourZ+QiU3wSRDorhWSh/p5IaWjMNPSyzpDiyKx6c/E/r5dgcOOmQsUJcsWWi4JEEozIPALiC80ZymlGKNMiu5WwEdWUYRbUPARn9eV10q5VHbvqPFxV6rU8jiKcwTlcggPXUId7aEALGGh4hld4sybWi/VufSxbC1Y+cwp/YH3+AK+Mkeg=</latexit><latexit sha1_base64="McN/C1V0wC+ZFgBefHyruFQAwvA=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoHgqePFYoV/QxrLZbNqlm03YnVhK6P/w4kERr/4Xb/4b0zYHbX0w8Hhvhpl5XiyFQdv+tgobm1vbO8Xd0t7+weFR+fikbaJEM95ikYx016OGS6F4CwVK3o01p6Eneccb3839zhPXRkSqidOYuyEdKhEIRjGTHtP+RPh8RJE0Z7elQbliV+0FyDpxclKBHI1B+avvRywJuUImqTE9x47RTalGwSSflfqJ4TFlYzrkvYwqGnLjpourZ+QiU3wSRDorhWSh/p5IaWjMNPSyzpDiyKx6c/E/r5dgcOOmQsUJcsWWi4JEEozIPALiC80ZymlGKNMiu5WwEdWUYRbUPARn9eV10q5VHbvqPFxV6rU8jiKcwTlcggPXUId7aEALGGh4hld4sybWi/VufSxbC1Y+cwp/YH3+AK+Mkeg=</latexit>

periodization operator

we have:


(4) periodization of the self-energy



0k

00k


point to any other.


A(!) ⌘ 1

Lc

X

r

A( eR,r)(!) =1

L

X

R

AR

(!) . (41)




kk



kk

0(!) 7! G

kk

(!)�k,k0 ⌘ bT [G]

kk



bT [G]

RR

0=

1

L

X

R

00R

000

�R�R

0,R00�R

000GR

00R

000 . (42)

For R = R


bT [G]

RR

(!) =1

L

X

R

00R

000

�R

00R

000GR

00R

000(!) =

1

L

X

R

GRR

(!) =1

Lc

X

r

Grr

(!) , (43)



⌃(!) 7! bT [⌃](!) . (44)



bT [⌃]( eR,r),( eR0,r0)(!) =1

Lc

X

r

00,r000

� eR+r� e

R

0�r

0,r00�r

000⌃r

00,r000(!) . (45)




0k

00k


point to any other.


A(!) ⌘ 1

Lc

X

r

A( eR,r)(!) =1

L

X

R

AR

(!) . (41)




kk



kk

0(!) 7! G

kk

(!)�k,k0 ⌘ bT [G]

kk



bT [G]

RR

0=

1

L

X

R

00R

000

�R�R

0,R00�R

000GR

00R

000 . (42)

For R = R


bT [G]

RR

(!) =1

L

X

R

00R

000

�R

00R

000GR

00R

000(!) =

1

L

X

R

GRR

(!) =1

Lc

X

r

Grr

(!) , (43)



⌃(!) 7! bT [⌃](!) . (44)



bT [⌃]( eR,r),( eR0,r0)(!) =1

Lc

X

r

00,r000

� eR+r� e

R

0�r

0,r00�r

000⌃r

00,r000(!) . (45)


G-periodization:

-periodization:⌃<latexit sha1_base64="37asvsx+S2aYtZmoZ+RYa/Sn0rE=">AAAB/XicdVBLSwMxGMzWV62v9XHzEiyCp7Lb+qi3ghePFe0DukvJZtM2NNksSVaoS/GvePGgiFf/hzf/jdl2BRUdCBlmvo9MJogZVdpxPqzCwuLS8kpxtbS2vrG5ZW/vtJVIJCYtLJiQ3QApwmhEWppqRrqxJIgHjHSC8UXmd26JVFREN3oSE5+jYUQHFCNtpL69l3qBYKGacHN513TI0bTUt8tOpV6v1k5q0Kk4MxjiOu75qQvdXCmDHM2+/e6FAiecRBozpFTPdWLtp0hqihmZlrxEkRjhMRqSnqER4kT56Sz9FB4aJYQDIc2JNJyp3zdSxFWWz0xypEfqt5eJf3m9RA/qfkqjONEkwvOHBgmDWsCsChhSSbBmE0MQltRkhXiEJMLaFJaV8PVT+D9pV00tFffquNyo5nUUwT44AEfABWegAS5BE7QABnfgATyBZ+veerRerNf5aMHKd3bBD1hvn+i5lXQ=</latexit><latexit sha1_base64="37asvsx+S2aYtZmoZ+RYa/Sn0rE=">AAAB/XicdVBLSwMxGMzWV62v9XHzEiyCp7Lb+qi3ghePFe0DukvJZtM2NNksSVaoS/GvePGgiFf/hzf/jdl2BRUdCBlmvo9MJogZVdpxPqzCwuLS8kpxtbS2vrG5ZW/vtJVIJCYtLJiQ3QApwmhEWppqRrqxJIgHjHSC8UXmd26JVFREN3oSE5+jYUQHFCNtpL69l3qBYKGacHN513TI0bTUt8tOpV6v1k5q0Kk4MxjiOu75qQvdXCmDHM2+/e6FAiecRBozpFTPdWLtp0hqihmZlrxEkRjhMRqSnqER4kT56Sz9FB4aJYQDIc2JNJyp3zdSxFWWz0xypEfqt5eJf3m9RA/qfkqjONEkwvOHBgmDWsCsChhSSbBmE0MQltRkhXiEJMLaFJaV8PVT+D9pV00tFffquNyo5nUUwT44AEfABWegAS5BE7QABnfgATyBZ+veerRerNf5aMHKd3bBD1hvn+i5lXQ=</latexit><latexit sha1_base64="37asvsx+S2aYtZmoZ+RYa/Sn0rE=">AAAB/XicdVBLSwMxGMzWV62v9XHzEiyCp7Lb+qi3ghePFe0DukvJZtM2NNksSVaoS/GvePGgiFf/hzf/jdl2BRUdCBlmvo9MJogZVdpxPqzCwuLS8kpxtbS2vrG5ZW/vtJVIJCYtLJiQ3QApwmhEWppqRrqxJIgHjHSC8UXmd26JVFREN3oSE5+jYUQHFCNtpL69l3qBYKGacHN513TI0bTUt8tOpV6v1k5q0Kk4MxjiOu75qQvdXCmDHM2+/e6FAiecRBozpFTPdWLtp0hqihmZlrxEkRjhMRqSnqER4kT56Sz9FB4aJYQDIc2JNJyp3zdSxFWWz0xypEfqt5eJf3m9RA/qfkqjONEkwvOHBgmDWsCsChhSSbBmE0MQltRkhXiEJMLaFJaV8PVT+D9pV00tFffquNyo5nUUwT44AEfABWegAS5BE7QABnfgATyBZ+veerRerNf5aMHKd3bBD1hvn+i5lXQ=</latexit><latexit sha1_base64="37asvsx+S2aYtZmoZ+RYa/Sn0rE=">AAAB/XicdVBLSwMxGMzWV62v9XHzEiyCp7Lb+qi3ghePFe0DukvJZtM2NNksSVaoS/GvePGgiFf/hzf/jdl2BRUdCBlmvo9MJogZVdpxPqzCwuLS8kpxtbS2vrG5ZW/vtJVIJCYtLJiQ3QApwmhEWppqRrqxJIgHjHSC8UXmd26JVFREN3oSE5+jYUQHFCNtpL69l3qBYKGacHN513TI0bTUt8tOpV6v1k5q0Kk4MxjiOu75qQvdXCmDHM2+/e6FAiecRBozpFTPdWLtp0hqihmZlrxEkRjhMRqSnqER4kT56Sz9FB4aJYQDIc2JNJyp3zdSxFWWz0xypEfqt5eJf3m9RA/qfkqjONEkwvOHBgmDWsCsChhSSbBmE0MQltRkhXiEJMLaFJaV8PVT+D9pV00tFffquNyo5nUUwT44AEfABWegAS5BE7QABnfgATyBZ+veerRerNf5aMHKd3bBD1hvn+i5lXQ=</latexit>

standard procedure, see CPT result

also ad hoc, but more artificial !as performed at an earlier stage

Lc=10

Lc=8

Lc=6

Lc=4 Lc=2Lc=1

both can be generalized to restore!the correct rotational (point-group) symmetries as well!






a<latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit><latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit><latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit><latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit>

H<latexit sha1_base64="nndXjcwcEIGbsQvoGgEJ57NyyR8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeClx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT+4XffUKleSwfzCxBP6JjyUPOqLFSqzEsV9yquwTZJF5OKpCjOSx/DUYxSyOUhgmqdd9zE+NnVBnOBM5Lg1RjQtmUjrFvqaQRaj9bHjonV1YZkTBWtqQhS/X3REYjrWdRYDsjaiZ63VuI/3n91IQ1P+MySQ1KtloUpoKYmCy+JiOukBkxs4Qyxe2thE2ooszYbEo2BG/95U3Sual6btVr3VbqtTyOIlzAJVyDB3dQhwY0oQ0MEJ7hFd6cR+fFeXc+Vq0FJ585hz9wPn8AmneMwg==</latexit><latexit sha1_base64="nndXjcwcEIGbsQvoGgEJ57NyyR8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeClx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT+4XffUKleSwfzCxBP6JjyUPOqLFSqzEsV9yquwTZJF5OKpCjOSx/DUYxSyOUhgmqdd9zE+NnVBnOBM5Lg1RjQtmUjrFvqaQRaj9bHjonV1YZkTBWtqQhS/X3REYjrWdRYDsjaiZ63VuI/3n91IQ1P+MySQ1KtloUpoKYmCy+JiOukBkxs4Qyxe2thE2ooszYbEo2BG/95U3Sual6btVr3VbqtTyOIlzAJVyDB3dQhwY0oQ0MEJ7hFd6cR+fFeXc+Vq0FJ585hz9wPn8AmneMwg==</latexit><latexit sha1_base64="nndXjcwcEIGbsQvoGgEJ57NyyR8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeClx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT+4XffUKleSwfzCxBP6JjyUPOqLFSqzEsV9yquwTZJF5OKpCjOSx/DUYxSyOUhgmqdd9zE+NnVBnOBM5Lg1RjQtmUjrFvqaQRaj9bHjonV1YZkTBWtqQhS/X3REYjrWdRYDsjaiZ63VuI/3n91IQ1P+MySQ1KtloUpoKYmCy+JiOukBkxs4Qyxe2thE2ooszYbEo2BG/95U3Sual6btVr3VbqtTyOIlzAJVyDB3dQhwY0oQ0MEJ7hFd6cR+fFeXc+Vq0FJ585hz9wPn8AmneMwg==</latexit><latexit sha1_base64="nndXjcwcEIGbsQvoGgEJ57NyyR8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeClx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT+4XffUKleSwfzCxBP6JjyUPOqLFSqzEsV9yquwTZJF5OKpCjOSx/DUYxSyOUhgmqdd9zE+NnVBnOBM5Lg1RjQtmUjrFvqaQRaj9bHjonV1YZkTBWtqQhS/X3REYjrWdRYDsjaiZ63VuI/3n91IQ1P+MySQ1KtloUpoKYmCy+JiOukBkxs4Qyxe2thE2ooszYbEo2BG/95U3Sual6btVr3VbqtTyOIlzAJVyDB3dQhwY0oQ0MEJ7hFd6cR+fFeXc+Vq0FJ585hz9wPn8AmneMwg==</latexit>

H 0<latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit><latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit><latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit><latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit>


H<latexit sha1_base64="TLXXT4Z2kyEo2fA9u+yRohUB6mY=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsyIYJcFN11WsA/oDCWTZtrQTDIkGWEY+htuXCji1p9x59+YaWehrQcCh3Pu4d6cMOFMG9f9dipb2zu7e9X92sHh0fFJ/fSsr2WqCO0RyaUahlhTzgTtGWY4HSaK4jjkdBDO7wt/8ESVZlI8miyhQYyngkWMYGMl35fWLLJ5ZzGuN9ymuwTaJF5JGlCiO65/+RNJ0pgKQzjWeuS5iQlyrAwjnC5qfqppgskcT+nIUoFjqoN8efMCXVllgiKp7BMGLdXfiRzHWmdxaCdjbGZ63SvE/7xRaqJWkDORpIYKsloUpRwZiYoC0IQpSgzPLMFEMXsrIjOsMDG2ppotwVv/8ibp3zQ9t+k93DbarbKOKlzAJVyDB3fQhg50oQcEEniGV3hzUufFeXc+VqMVp8ycwx84nz9ziJHo</latexit><latexit sha1_base64="TLXXT4Z2kyEo2fA9u+yRohUB6mY=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsyIYJcFN11WsA/oDCWTZtrQTDIkGWEY+htuXCji1p9x59+YaWehrQcCh3Pu4d6cMOFMG9f9dipb2zu7e9X92sHh0fFJ/fSsr2WqCO0RyaUahlhTzgTtGWY4HSaK4jjkdBDO7wt/8ESVZlI8miyhQYyngkWMYGMl35fWLLJ5ZzGuN9ymuwTaJF5JGlCiO65/+RNJ0pgKQzjWeuS5iQlyrAwjnC5qfqppgskcT+nIUoFjqoN8efMCXVllgiKp7BMGLdXfiRzHWmdxaCdjbGZ63SvE/7xRaqJWkDORpIYKsloUpRwZiYoC0IQpSgzPLMFEMXsrIjOsMDG2ppotwVv/8ibp3zQ9t+k93DbarbKOKlzAJVyDB3fQhg50oQcEEniGV3hzUufFeXc+VqMVp8ycwx84nz9ziJHo</latexit><latexit sha1_base64="TLXXT4Z2kyEo2fA9u+yRohUB6mY=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsyIYJcFN11WsA/oDCWTZtrQTDIkGWEY+htuXCji1p9x59+YaWehrQcCh3Pu4d6cMOFMG9f9dipb2zu7e9X92sHh0fFJ/fSsr2WqCO0RyaUahlhTzgTtGWY4HSaK4jjkdBDO7wt/8ESVZlI8miyhQYyngkWMYGMl35fWLLJ5ZzGuN9ymuwTaJF5JGlCiO65/+RNJ0pgKQzjWeuS5iQlyrAwjnC5qfqppgskcT+nIUoFjqoN8efMCXVllgiKp7BMGLdXfiRzHWmdxaCdjbGZ63SvE/7xRaqJWkDORpIYKsloUpRwZiYoC0IQpSgzPLMFEMXsrIjOsMDG2ppotwVv/8ibp3zQ9t+k93DbarbKOKlzAJVyDB3fQhg50oQcEEniGV3hzUufFeXc+VqMVp8ycwx84nz9ziJHo</latexit><latexit sha1_base64="TLXXT4Z2kyEo2fA9u+yRohUB6mY=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsyIYJcFN11WsA/oDCWTZtrQTDIkGWEY+htuXCji1p9x59+YaWehrQcCh3Pu4d6cMOFMG9f9dipb2zu7e9X92sHh0fFJ/fSsr2WqCO0RyaUahlhTzgTtGWY4HSaK4jjkdBDO7wt/8ESVZlI8miyhQYyngkWMYGMl35fWLLJ5ZzGuN9ymuwTaJF5JGlCiO65/+RNJ0pgKQzjWeuS5iQlyrAwjnC5qfqppgskcT+nIUoFjqoN8efMCXVllgiKp7BMGLdXfiRzHWmdxaCdjbGZ63SvE/7xRaqJWkDORpIYKsloUpRwZiYoC0IQpSgzPLMFEMXsrIjOsMDG2ppotwVv/8ibp3zQ9t+k93DbarbKOKlzAJVyDB3fQhg50oQcEEniGV3hzUufFeXc+VqMVp8ycwx84nz9ziJHo</latexit>

a)<latexit sha1_base64="M5JmkpsWrz4R8opay4sS7CLjOBU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3o1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASPTjQ4=</latexit><latexit sha1_base64="M5JmkpsWrz4R8opay4sS7CLjOBU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3o1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASPTjQ4=</latexit><latexit sha1_base64="M5JmkpsWrz4R8opay4sS7CLjOBU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3o1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASPTjQ4=</latexit><latexit sha1_base64="M5JmkpsWrz4R8opay4sS7CLjOBU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3o1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASPTjQ4=</latexit>

b)<latexit sha1_base64="ND4qzIWQ/A8rLJ9FcKjhRw08rko=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPQRXg3LFrboLkHXi5aQCOZqD8ld/GLM04gqZpMb0PDdBP6MaBZN8VuqnhieUTeiI9yxVNOLGzxaXzsiFVYYkjLUthWSh/p7IaGTMNApsZ0RxbFa9ufif10sxrPuZUEmKXLHlojCVBGMyf5sMheYM5dQSyrSwtxI2ppoytOGUbAje6svrpH1d9dyqd1+rNOp5HEU4g3O4BA9uoAF30IQWMAjhGV7hzZk4L86787FsLTj5zCn8gfP5AyVYjQ8=</latexit><latexit sha1_base64="ND4qzIWQ/A8rLJ9FcKjhRw08rko=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPQRXg3LFrboLkHXi5aQCOZqD8ld/GLM04gqZpMb0PDdBP6MaBZN8VuqnhieUTeiI9yxVNOLGzxaXzsiFVYYkjLUthWSh/p7IaGTMNApsZ0RxbFa9ufif10sxrPuZUEmKXLHlojCVBGMyf5sMheYM5dQSyrSwtxI2ppoytOGUbAje6svrpH1d9dyqd1+rNOp5HEU4g3O4BA9uoAF30IQWMAjhGV7hzZk4L86787FsLTj5zCn8gfP5AyVYjQ8=</latexit><latexit sha1_base64="ND4qzIWQ/A8rLJ9FcKjhRw08rko=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPQRXg3LFrboLkHXi5aQCOZqD8ld/GLM04gqZpMb0PDdBP6MaBZN8VuqnhieUTeiI9yxVNOLGzxaXzsiFVYYkjLUthWSh/p7IaGTMNApsZ0RxbFa9ufif10sxrPuZUEmKXLHlojCVBGMyf5sMheYM5dQSyrSwtxI2ppoytOGUbAje6svrpH1d9dyqd1+rNOp5HEU4g3O4BA9uoAF30IQWMAjhGV7hzZk4L86787FsLTj5zCn8gfP5AyVYjQ8=</latexit><latexit sha1_base64="ND4qzIWQ/A8rLJ9FcKjhRw08rko=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPQRXg3LFrboLkHXi5aQCOZqD8ld/GLM04gqZpMb0PDdBP6MaBZN8VuqnhieUTeiI9yxVNOLGzxaXzsiFVYYkjLUthWSh/p7IaGTMNApsZ0RxbFa9ufif10sxrPuZUEmKXLHlojCVBGMyf5sMheYM5dQSyrSwtxI2ppoytOGUbAje6svrpH1d9dyqd1+rNOp5HEU4g3O4BA9uoAF30IQWMAjhGV7hzZk4L86787FsLTj5zCn8gfP5AyVYjQ8=</latexit>

c)<latexit sha1_base64="5Z2N0CXj76wDVzOIujoY2082LW4=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3Y1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASbdjRA=</latexit><latexit sha1_base64="5Z2N0CXj76wDVzOIujoY2082LW4=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3Y1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASbdjRA=</latexit><latexit sha1_base64="5Z2N0CXj76wDVzOIujoY2082LW4=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3Y1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASbdjRA=</latexit><latexit sha1_base64="5Z2N0CXj76wDVzOIujoY2082LW4=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBZBLyWRgj0WvHisYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akmt3O/84RK81g+mmmCfkRHkoecUWOlB3Y1KFfcqrsAWSdeTiqQozkof/WHMUsjlIYJqnXPcxPjZ1QZzgTOSv1UY0LZhI6wZ6mkEWo/W1w6IxdWGZIwVrakIQv190RGI62nUWA7I2rGetWbi/95vdSEdT/jMkkNSrZcFKaCmJjM3yZDrpAZMbWEMsXtrYSNqaLM2HBKNgRv9eV10r6uem7Vu69VGvU8jiKcwTlcggc30IA7aEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8ASbdjRA=</latexit>

d)<latexit sha1_base64="nQkmdRzL3CwaeniztwLGiUcYXCg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi2GPBi8cq9gPaUDabSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSAXXxnW/nbX1jc2t7dJOeXdv/+CwcnTc1kmmGLZYIhLVDahGwSW2DDcCu6lCGgcCO8H4duZ3nlBpnshHM0nRj+lQ8ogzaqz0EF4OKlW35s5BVolXkCoUaA4qX/0wYVmM0jBBte55bmr8nCrDmcBpuZ9pTCkb0yH2LJU0Ru3n80un5NwqIYkSZUsaMld/T+Q01noSB7Yzpmakl72Z+J/Xy0xU93Mu08ygZItFUSaIScjsbRJyhcyIiSWUKW5vJWxEFWXGhlO2IXjLL6+S9lXNc2ve/XW1US/iKMEpnMEFeHADDbiDJrSAQQTP8Apvzth5cd6dj0XrmlPMnMAfOJ8/KGKNEQ==</latexit><latexit sha1_base64="nQkmdRzL3CwaeniztwLGiUcYXCg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi2GPBi8cq9gPaUDabSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSAXXxnW/nbX1jc2t7dJOeXdv/+CwcnTc1kmmGLZYIhLVDahGwSW2DDcCu6lCGgcCO8H4duZ3nlBpnshHM0nRj+lQ8ogzaqz0EF4OKlW35s5BVolXkCoUaA4qX/0wYVmM0jBBte55bmr8nCrDmcBpuZ9pTCkb0yH2LJU0Ru3n80un5NwqIYkSZUsaMld/T+Q01noSB7Yzpmakl72Z+J/Xy0xU93Mu08ygZItFUSaIScjsbRJyhcyIiSWUKW5vJWxEFWXGhlO2IXjLL6+S9lXNc2ve/XW1US/iKMEpnMEFeHADDbiDJrSAQQTP8Apvzth5cd6dj0XrmlPMnMAfOJ8/KGKNEQ==</latexit><latexit sha1_base64="nQkmdRzL3CwaeniztwLGiUcYXCg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi2GPBi8cq9gPaUDabSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSAXXxnW/nbX1jc2t7dJOeXdv/+CwcnTc1kmmGLZYIhLVDahGwSW2DDcCu6lCGgcCO8H4duZ3nlBpnshHM0nRj+lQ8ogzaqz0EF4OKlW35s5BVolXkCoUaA4qX/0wYVmM0jBBte55bmr8nCrDmcBpuZ9pTCkb0yH2LJU0Ru3n80un5NwqIYkSZUsaMld/T+Q01noSB7Yzpmakl72Z+J/Xy0xU93Mu08ygZItFUSaIScjsbRJyhcyIiSWUKW5vJWxEFWXGhlO2IXjLL6+S9lXNc2ve/XW1US/iKMEpnMEFeHADDbiDJrSAQQTP8Apvzth5cd6dj0XrmlPMnMAfOJ8/KGKNEQ==</latexit><latexit sha1_base64="nQkmdRzL3CwaeniztwLGiUcYXCg=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi2GPBi8cq9gPaUDabSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSAXXxnW/nbX1jc2t7dJOeXdv/+CwcnTc1kmmGLZYIhLVDahGwSW2DDcCu6lCGgcCO8H4duZ3nlBpnshHM0nRj+lQ8ogzaqz0EF4OKlW35s5BVolXkCoUaA4qX/0wYVmM0jBBte55bmr8nCrDmcBpuZ9pTCkb0yH2LJU0Ru3n80un5NwqIYkSZUsaMld/T+Q01noSB7Yzpmakl72Z+J/Xy0xU93Mu08ygZItFUSaIScjsbRJyhcyIiSWUKW5vJWxEFWXGhlO2IXjLL6+S9lXNc2ve/XW1US/iKMEpnMEFeHADDbiDJrSAQQTP8Apvzth5cd6dj0XrmlPMnMAfOJ8/KGKNEQ==</latexit>


�t<latexit sha1_base64="Z2mqNenuVkwN4tiMdi8WpLocTcQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPVzhoFxxq+4CZJ14OalAjuag/NUfxiyNuEImqTE9z03Qz6hGwSSflfqp4QllEzriPUsVjbjxs8WlM3JhlSEJY21LIVmovycyGhkzjQLbGVEcm1VvLv7n9VIM634mVJIiV2y5KEwlwZjM3yZDoTlDObWEMi3srYSNqaYMbTglG4K3+vI6aV9XPbfq3dcqjXoeRxHO4BwuwYMbaMAdNKEFDEJ4hld4cybOi/PufCxbC04+cwp/4Hz+AEZ7jSU=</latexit><latexit sha1_base64="Z2mqNenuVkwN4tiMdi8WpLocTcQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPVzhoFxxq+4CZJ14OalAjuag/NUfxiyNuEImqTE9z03Qz6hGwSSflfqp4QllEzriPUsVjbjxs8WlM3JhlSEJY21LIVmovycyGhkzjQLbGVEcm1VvLv7n9VIM634mVJIiV2y5KEwlwZjM3yZDoTlDObWEMi3srYSNqaYMbTglG4K3+vI6aV9XPbfq3dcqjXoeRxHO4BwuwYMbaMAdNKEFDEJ4hld4cybOi/PufCxbC04+cwp/4Hz+AEZ7jSU=</latexit><latexit sha1_base64="Z2mqNenuVkwN4tiMdi8WpLocTcQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPVzhoFxxq+4CZJ14OalAjuag/NUfxiyNuEImqTE9z03Qz6hGwSSflfqp4QllEzriPUsVjbjxs8WlM3JhlSEJY21LIVmovycyGhkzjQLbGVEcm1VvLv7n9VIM634mVJIiV2y5KEwlwZjM3yZDoTlDObWEMi3srYSNqaYMbTglG4K3+vI6aV9XPbfq3dcqjXoeRxHO4BwuwYMbaMAdNKEFDEJ4hld4cybOi/PufCxbC04+cwp/4Hz+AEZ7jSU=</latexit><latexit sha1_base64="Z2mqNenuVkwN4tiMdi8WpLocTcQ=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyWRgj0WvHisYj+gDWWz3bRLN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkQKg6777RQ2Nre2d4q7pb39g8Oj8vFJ28SpZrzFYhnrbkANl0LxFgqUvJtoTqNA8k4wuZ37nSeujYjVI04T7kd0pEQoGEUrPVzhoFxxq+4CZJ14OalAjuag/NUfxiyNuEImqTE9z03Qz6hGwSSflfqp4QllEzriPUsVjbjxs8WlM3JhlSEJY21LIVmovycyGhkzjQLbGVEcm1VvLv7n9VIM634mVJIiV2y5KEwlwZjM3yZDoTlDObWEMi3srYSNqaYMbTglG4K3+vI6aV9XPbfq3dcqjXoeRxHO4BwuwYMbaMAdNKEFDEJ4hld4cybOi/PufCxbC04+cwp/4Hz+AEZ7jSU=</latexit>

U<latexit sha1_base64="Kd2yNkybdNWhIfKB0PbbMtRy6bs=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokI9ljw4rEF0xbaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LBTFMMYjqSPOKMGiu1/EGl6tbcBcg68QpShQLNQeWrP0xYFqM0TFCte56bmiCnynAmcFbuZxpTyiZ0hD1LJY1RB/ni0Bm5tMqQRImyJQ1ZqL8nchprPY1D2xlTM9ar3lz8z+tlJqoHOZdpZlCy5aIoE8QkZP41GXKFzIipJZQpbm8lbEwVZcZmU7YheKsvr5P2dc1za17rptqoF3GU4Bwu4Ao8uIUG3EMTfGCA8Ayv8OY8Oi/Ou/OxbN1wipkz+APn8weuK4zP</latexit><latexit sha1_base64="Kd2yNkybdNWhIfKB0PbbMtRy6bs=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokI9ljw4rEF0xbaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LBTFMMYjqSPOKMGiu1/EGl6tbcBcg68QpShQLNQeWrP0xYFqM0TFCte56bmiCnynAmcFbuZxpTyiZ0hD1LJY1RB/ni0Bm5tMqQRImyJQ1ZqL8nchprPY1D2xlTM9ar3lz8z+tlJqoHOZdpZlCy5aIoE8QkZP41GXKFzIipJZQpbm8lbEwVZcZmU7YheKsvr5P2dc1za17rptqoF3GU4Bwu4Ao8uIUG3EMTfGCA8Ayv8OY8Oi/Ou/OxbN1wipkz+APn8weuK4zP</latexit><latexit sha1_base64="Kd2yNkybdNWhIfKB0PbbMtRy6bs=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokI9ljw4rEF0xbaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LBTFMMYjqSPOKMGiu1/EGl6tbcBcg68QpShQLNQeWrP0xYFqM0TFCte56bmiCnynAmcFbuZxpTyiZ0hD1LJY1RB/ni0Bm5tMqQRImyJQ1ZqL8nchprPY1D2xlTM9ar3lz8z+tlJqoHOZdpZlCy5aIoE8QkZP41GXKFzIipJZQpbm8lbEwVZcZmU7YheKsvr5P2dc1za17rptqoF3GU4Bwu4Ao8uIUG3EMTfGCA8Ayv8OY8Oi/Ou/OxbN1wipkz+APn8weuK4zP</latexit><latexit sha1_base64="Kd2yNkybdNWhIfKB0PbbMtRy6bs=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokI9ljw4rEF0xbaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LBTFMMYjqSPOKMGiu1/EGl6tbcBcg68QpShQLNQeWrP0xYFqM0TFCte56bmiCnynAmcFbuZxpTyiZ0hD1LJY1RB/ni0Bm5tMqQRImyJQ1ZqL8nchprPY1D2xlTM9ar3lz8z+tlJqoHOZdpZlCy5aIoE8QkZP41GXKFzIipJZQpbm8lbEwVZcZmU7YheKsvr5P2dc1za17rptqoF3GU4Bwu4Ao8uIUG3EMTfGCA8Ayv8OY8Oi/Ou/OxbN1wipkz+APn8weuK4zP</latexit>

Tran Minh-Tien (2006)Biroli et al. (2004)

(5) periodic CPTt’: disconnected clusters with periodic b.c. !modify the parameters of H rather than H’:!

t 7! t<latexit sha1_base64="31MKrjb8/6YWmQYy7TKVlD7PPrY=">AAACGXicbVDLSsNAFJ34rPEVdelmsAiuSlIEXRbcuKxgH9CEMplM2qHzCDMToYT+hht/xY0LRVzqyr9x0mahrReGOZx7z32cOGNUG9//dtbWNza3tms77u7e/sGhd3Tc1TJXmHSwZFL1Y6QJo4J0DDWM9DNFEI8Z6cWTmzLfeyBKUynuzTQjEUcjQVOKkbHU0POLMJYs0VNuP2hmbshRpo10Q2llZdelgqFX9xv+POAqCCpQB1W0h95nmEiccyIMZkjrQeBnJiqQMhQzYgfmmmQIT9CIDCwUiBMdFfPLZvDcMglMpbJPGDhnfysKxHW5mq3kyIz1cq4k/8sNcpNeRwUVWW6IwItBaW4PlLC0CSZUEWzY1AKEFbW7QjxGCmFjzXStCcHyyaug22wEfiO4u6y3mpUdNXAKzsAFCMAVaIFb0AYdgMEjeAav4M15cl6cd+djUbrmVJoT8Cecrx8DOaF9</latexit><latexit sha1_base64="31MKrjb8/6YWmQYy7TKVlD7PPrY=">AAACGXicbVDLSsNAFJ34rPEVdelmsAiuSlIEXRbcuKxgH9CEMplM2qHzCDMToYT+hht/xY0LRVzqyr9x0mahrReGOZx7z32cOGNUG9//dtbWNza3tms77u7e/sGhd3Tc1TJXmHSwZFL1Y6QJo4J0DDWM9DNFEI8Z6cWTmzLfeyBKUynuzTQjEUcjQVOKkbHU0POLMJYs0VNuP2hmbshRpo10Q2llZdelgqFX9xv+POAqCCpQB1W0h95nmEiccyIMZkjrQeBnJiqQMhQzYgfmmmQIT9CIDCwUiBMdFfPLZvDcMglMpbJPGDhnfysKxHW5mq3kyIz1cq4k/8sNcpNeRwUVWW6IwItBaW4PlLC0CSZUEWzY1AKEFbW7QjxGCmFjzXStCcHyyaug22wEfiO4u6y3mpUdNXAKzsAFCMAVaIFb0AYdgMEjeAav4M15cl6cd+djUbrmVJoT8Cecrx8DOaF9</latexit><latexit sha1_base64="31MKrjb8/6YWmQYy7TKVlD7PPrY=">AAACGXicbVDLSsNAFJ34rPEVdelmsAiuSlIEXRbcuKxgH9CEMplM2qHzCDMToYT+hht/xY0LRVzqyr9x0mahrReGOZx7z32cOGNUG9//dtbWNza3tms77u7e/sGhd3Tc1TJXmHSwZFL1Y6QJo4J0DDWM9DNFEI8Z6cWTmzLfeyBKUynuzTQjEUcjQVOKkbHU0POLMJYs0VNuP2hmbshRpo10Q2llZdelgqFX9xv+POAqCCpQB1W0h95nmEiccyIMZkjrQeBnJiqQMhQzYgfmmmQIT9CIDCwUiBMdFfPLZvDcMglMpbJPGDhnfysKxHW5mq3kyIz1cq4k/8sNcpNeRwUVWW6IwItBaW4PlLC0CSZUEWzY1AKEFbW7QjxGCmFjzXStCcHyyaug22wEfiO4u6y3mpUdNXAKzsAFCMAVaIFb0AYdgMEjeAav4M15cl6cd+djUbrmVJoT8Cecrx8DOaF9</latexit><latexit sha1_base64="31MKrjb8/6YWmQYy7TKVlD7PPrY=">AAACGXicbVDLSsNAFJ34rPEVdelmsAiuSlIEXRbcuKxgH9CEMplM2qHzCDMToYT+hht/xY0LRVzqyr9x0mahrReGOZx7z32cOGNUG9//dtbWNza3tms77u7e/sGhd3Tc1TJXmHSwZFL1Y6QJo4J0DDWM9DNFEI8Z6cWTmzLfeyBKUynuzTQjEUcjQVOKkbHU0POLMJYs0VNuP2hmbshRpo10Q2llZdelgqFX9xv+POAqCCpQB1W0h95nmEiccyIMZkjrQeBnJiqQMhQzYgfmmmQIT9CIDCwUiBMdFfPLZvDcMglMpbJPGDhnfysKxHW5mq3kyIz1cq4k/8sNcpNeRwUVWW6IwItBaW4PlLC0CSZUEWzY1AKEFbW7QjxGCmFjzXStCcHyyaug22wEfiO4u6y3mpUdNXAKzsAFCMAVaIFb0AYdgMEjeAav4M15cl6cd+djUbrmVJoT8Cecrx8DOaF9</latexit>

• such that the symmetries ofH and H’ are the same !

• superlattice translations: !• cluster translations: X!• add the necessary hopping

parameters!• irrelevant for !!• explicitly:

X<latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit><latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit><latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit><latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit>

Lc ! 1<latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit><latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit><latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit><latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit>

t = (V W )U †tU(V W )†<latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit><latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit><latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit><latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit>


(5) periodic CPTt’: disconnected clusters with periodic b.c. !modify the parameters of H rather than H’:!


• such that the symmetries ofH and H’ are the same !

• superlattice translations: !• cluster translations: X!• add the necessary hopping

parameters!• irrelevant for !!• explicitly:

X<latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit><latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit><latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit><latexit sha1_base64="NHKt44AfFMm7b6PgzGx4t9uwvrM=">AAAB8nicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHgxWME84DNEmYns8mQeSwzs0JY8hlePCji1a/x5t84m+xBEwsaiqpuurvilDNjff/b29jc2t7ZrexV9w8Oj45rJ6ddozJNaIcornQ/xoZyJmnHMstpP9UUi5jTXjy9K/zeE9WGKfloZymNBB5LljCCrZPCAZlQMhVYT6vDWt1v+AugdRKUpA4l2sPa12CkSCaotIRjY8LAT22UY20Z4XReHWSGpphM8ZiGjkosqInyxclzdOmUEUqUdiUtWqi/J3IsjJmJ2HUKbCdm1SvE/7wws8ltlDOZZpZKslyUZBxZhYr/0YhpSiyfOYKJZu5WRCZYY2JdSkUIwerL66TbbAR+I3i4rreaZRwVOIcLuIIAbqAF99CGDhBQ8Ayv8OZZ78V79z6WrRteOXMGf+B9/gDZlZDn</latexit>

Lc ! 1<latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit><latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit><latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit><latexit sha1_base64="S68LAZubSOSPs4W/3uHK5ft8cYc=">AAAB83icbVBNS8NAEJ3Ur1q/qh69BIvgqSRF0GPBiwcPFewHNKFstpt26WY37E6EEvo3vHhQxKt/xpv/xm2bg7Y+GHi8N8PMvCgV3KDnfTuljc2t7Z3ybmVv/+DwqHp80jEq05S1qRJK9yJimOCStZGjYL1UM5JEgnWjye3c7z4xbbiSjzhNWZiQkeQxpwStFNwPaIAq4DLG6aBa8+reAu468QtSgwKtQfUrGCqaJUwiFcSYvu+lGOZEI6eCzSpBZlhK6ISMWN9SSRJmwnxx88y9sMrQjZW2JdFdqL8ncpIYM00i25kQHJtVby7+5/UzjG/CnMs0QybpclGcCReVOw/AHXLNKIqpJYRqbm916ZhoQtHGVLEh+Ksvr5NOo+57df/hqtZsFHGU4QzO4RJ8uIYm3EEL2kAhhWd4hTcnc16cd+dj2VpyiplT+APn8wcrK5G1</latexit>

t = (V W )U †tU(V W )†<latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit><latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit><latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit><latexit sha1_base64="aFbtHl3gxDuWQpnUZNF5AwF4XBY=">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</latexit>


















eR

<latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit><latexit sha1_base64="TqD3qifs+xbdwt5e24YKTk034OA=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiQi2GXBjcsq9gFNKJPJTTt0MgkzE6WEgBt/xY0LRdz6E+78GydtFtp6YJjDOfdy7z1+wqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaLvYwmMcugoqhj0EwE48hn0/MlV4ffuQUga8zs1TcCL8IjTkBKstDQ0j9wHGoCiLIDM9WMWyGmkv+w2z4dm3W7YM1jLxClJHZVoD80vN4hJGgFXhGEpB46dKC/DQlHCIK+5qYQEkwkewUBTjiOQXja7IbdOtRJYYSz048qaqb87MhzJYjddGWE1loteIf7nDVIVNr2M8iRVwMl8UJgyS8VWEYgVUAFEsakmmAiqd7XIGAtMlI6tpkNwFk9eJt3zhmM3nJuLeqtZxlFFx+gEnSEHXaIWukZt1EEEPaJn9IrejCfjxXg3PualFaPsOUR/YHz+AFhBmJE=</latexit>



r

<latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit><latexit sha1_base64="4bpQxqif47nSh28gZ57k4CA+5jk=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F6Zmo+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifP0cskvk=</latexit>

a

<latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit><latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit><latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit><latexit sha1_base64="ybexmHpkvRblzM2QSc2MW8mUXwI=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyI0C4LblxWsA9ox5LJZNrQTDIkGaUM/Q83LhRx67+482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT7papkqQjtEcqn6AdaUM0E7hhlO+4miOA447QXTm9zvPVKlmRT3ZpZQP8ZjwSJGsLHSwzCQPNSz2F4Zno+qNbfuLoDWiVeQGhRoj6pfw1CSNKbCEI61HnhuYvwMK8MIp/PKMNU0wWSKx3RgqcAx1X62SD1HF1YJUSSVPcKghfp7I8OxzqPZyRibiV71cvE/b5CaqOlnTCSpoYIsH4pSjoxEeQUoZIoSw2eWYKKYzYrIBCtMjC2qYkvwVr+8TrpXdc+te3fXtVazqKMMZ3AOl+BBA1pwC23oAAEFz/AKb86T8+K8Ox/L0ZJT7JzCHzifPy1Xkug=</latexit>


H �<latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit><latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit><latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit><latexit sha1_base64="0nl4gSodhuR+K1HD9ZUeET9PN/Y=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lEsMeClx6r2A9oQ9lsN+3SzSbsToQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7uZ+54lrI2L1iNOE+xEdKREKRtFKD43LQbniVt0FyDrxclKBHM1B+as/jFkacYVMUmN6npugn1GNgkk+K/VTwxPKJnTEe5YqGnHjZ4tLZ+TCKkMSxtqWQrJQf09kNDJmGgW2M6I4NqveXPzP66UY1vxMqCRFrthyUZhKgjGZv02GQnOGcmoJZVrYWwkbU00Z2nBKNgRv9eV10r6uem7Vu7+p1Gt5HEU4g3O4Ag9uoQ4NaEILGITwDK/w5kycF+fd+Vi2Fpx85hT+wPn8Afq/jPM=</latexit>

R = eR + r

<latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit><latexit sha1_base64="xwd37XW5fnGFZCO1+c+8qt4fIB4=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUBIR7EYsuHFZxT6gCWUyuW2HTh7MTJQS8jlu/BU3Iop065c4abOwrQeGOZxzL/fe48WcSWVZE2NldW19Y7O0Vd7e2d3bNw8OWzJKBIUmjXgkOh6RwFkITcUUh04sgAQeh7Y3us399hMIyaLwUY1jcAMyCFmfUaK01DNvUseLuC/Hgf7ShyzD184z80Ex7sOSd47nJJFlPbNiVa0p8DKxC1JBBRo988PxI5oEECrKiZRd24qVmxKhGOWQlZ1EQkzoiAygq2lIApBuOj00w6da8XE/EvqFCk/Vvx0pCWS+m64MiBrKRS8X//O6ierX3JSFcaIgpLNB/YRjFeE8NewzAVTxsSaECqZ3xXRIBKFKZ1vWIdiLJy+T1kXVtqr2/WWlXiviKKFjdILOkI2uUB3doQZqIope0Bv6RF/Gq/FufBuTWemKUfQcoTkYP79il6iV</latexit>









Fig. 9: Construction of the modified original system H for a one-dimensional lattice. a) Orig-inal system H , black lines: nearest-neighbor hopping �t, on-site Hubbard interaction U , unitcell C. b) Reference system H 0: superlattice (basis) vector e

R, cluster vector r, supercell SCwith Lc = 3 (cf. Fig. 2). c) A cyclic permutation of the sites within each cluster is a symmetryof H 0. Black lines in the upper panel: nearest-neighbor hopping parameters related to a givencluster. Lower panel: resulting hopping parameters after the transformation. d) Hopping pa-rameters of the Hamiltonian H , which is invariant (i) under all intra-cluster translantions and(ii) under all superlattice translations.

in t, while the lower one shows those that are generated by the translation. The hopping ampli-tudes generated by all cluster translations are included in the new hopping matrix t, see panel(d).

Note that for Lc ! 1 the inclusion of the additional hopping amplitudes becomes irrelevant.Hence, the replacement t 7! t is controlled by Lc, i.e., it becomes exact, up to irrelavantboundary terms, in the infinite-cluster limit. The explicit construction of t is the following:

t = (V W )U

†t U (V W )

† . (46)

Let us stress once more, that V W 6= U for clusters of finite size Lc. The first transformationU

†tU diagonalizes the hopping matrix t, and the diagonal elements are given by "(k) = "( e

G+

ek) with a uniquely defined reciprocal superlattice vector e

G and a uniquely defined wave vectorek 2 RSC. Double back transformation using V and W then yields:

tRR

0= t e

R,r; eR0,r0 =1

Lc

X

eG

ei eG(r�r

0)Lc

L

X

ek

eiek( eR� eR

0)"(ek +

eG) . (47)

"(k) =1

L

X

RR0

e�ik(R�R0)tRR0

<latexit sha1_base64="ZiOVsFLRvRWLGcWZPB0ikdcF0K4=">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</latexit><latexit sha1_base64="ZiOVsFLRvRWLGcWZPB0ikdcF0K4=">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</latexit><latexit sha1_base64="ZiOVsFLRvRWLGcWZPB0ikdcF0K4=">AAACm3icjVFNTxsxEPUupaUGSqAXpKqSRYQIB6JdVAkuICQOoCqHFBFAyobI68wGK157ZXuRImv/FD+lt/6bOiGHfPTQkSw/v5k3M55JC8GNjaI/Qbj2Yf3jp43PeHNr+8tObXfvwahSM+gwJZR+SqkBwSV0LLcCngoNNE8FPKaj64n/8RW04Ure23EBvZwOJc84o9ZT/dpb8ko1FIYLJUkjSZUYmHHuLzeqjvEFTjJNmYsr16oSU+Z9h+dj7qql5xGuMDy7E44XU+HGsu5kRXnstfY/CvRr9agZTY2sgngG6mhm7X7tdzJQrMxBWiaoMd04KmzPUW05E+ArlAYKykZ0CF0PJc3B9Nx0thU59MyAZEr7Iy2ZsvMKR3Mz6c5H5tS+mGXfhPyXr1va7LznuCxKC5K9F8pKQawik0WRAdfArBh7QJnmvlfCXqjfhvXrxH4I8fKXV8HDaTOOmvGvH/Wr09k4NtA3dIAaKEZn6ArdojbqIBbsB5fBTXAbfg+vw59h6z00DGaar2jBws5fvA/MJQ==</latexit><latexit sha1_base64="ZiOVsFLRvRWLGcWZPB0ikdcF0K4=">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</latexit>


(5) periodic CPT, contd.t’: disconnected clusters with periodic b.c. !modify the parameters of H rather than H’:!


• apply CPT to using ref.sys.!

• this is diagonalized by VW:!

• implicit periodization, identifying!

• self-energy is discontinuous:

H<latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit><latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit><latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit><latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit> H 0

<latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit><latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit><latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit><latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit>


Obviously, t is invariant under both, superlattice translations as well as under cluster transla-tions.

Applying CPT to the original system Fig. 9(d) using the reference system Fig. 9(b) yields

G(!) =1

! + µ � t + ⌃(!). (48)

As the self-energy is taken from the reference system, it is diagonal with respect to superlatticetranslations e

R and is eR-independent. Since the original as well as the reference system share

the same symmetries both, ⌃(!) and t are diagonalized by V W :

G(

ek +

eG,!) =

1

! + µ � "(ek +

eG) +⌃(

eG,!)

. (49)

Note that we have replaced the diagonal elements "(ek, eG) of (V W )

†t(V W ) by the diagonal

elements "(ek+ eG) of U

†tU , which becomes correct for Lc ! 1 as discussed above. Likewise,

again using the unique decomposition k =

ek +

eG, we interprete G(

ek +

eG,!) ⌘ G(

ek, e

G,!)

as G(k,!), i.e., as the diagonal elements of U

†G(!)U . In this way we have achieved our goal,

and we get a CPT Green’s function respecting the full translational symmetries of the originallattice. Obviously, the periodized CPT (P-CPT) employs an implicit periodization scheme,which is approximate for any finite Lc but becomes exact, as the CPT approximation itself, in theLc ! 1 limit. We note that for finite Lc, the k-dependence of the self-energy is discontinuous:⌃(

ek +

eG) = ⌃(

eG) is e

k-independent, i.e., within each of the Lc “patches” RSC of the fullreciprocal unit cell RC of the original lattice, the self-energy is constant, while it jumps whencrossing the boundaries between the patches. This unphysical feature of the P-CPT must betolerated.

8 Self-consistent cluster-embedding approaches

A big disadvantage of the CPT and related cluster approximations is the lacking self-consistency.Let us recall that a generic mean-field theory not only takes a local perspective and focusses ona single site or on a small cluster cut out of an infinite lattice but also aims at a proper embed-ding of the site or the cluster in a mean field. The purpose of this mean field is to approximatelyaccount for the neglected environment of the cluster. The mean-field feeds back to the clusterproblem but beyond that the quality of the mean-field theory improves considerably when thecluster observables feed back to the the mean field as well, i.e., when cluster observables andmean field are determined self-consistently.

Table 2 gives an overview over various cluster approximations without (first column) and withself-consistent cluster embedding. Taking the self-energy of a single Hubbard site (Lc = 1),

⌃RR

0�(!) = �RR

0UhnR��i + �

RR

0U2hn

R��i(1 � hnR��i)

! + µ � U(1 � hnR��i)

, (50)




G(!) =1

! + µ � t + ⌃(!). (48)




G(

ek +

eG,!) =

1

! + µ � "(ek +

eG) +⌃(

eG,!)

. (49)






ek +


ek +

eG,!) ⌘ G(

ek, e

G,!)




ek +

eG) = ⌃(

eG) is e





⌃RR

0�(!) = �RR

0UhnR��i + �

RR

0U2hn

R��i(1 � hnR��i)

! + µ � U(1 � hnR��i)

, (50)

G(ek, eG,!) ⌘ G(k,!)<latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit><latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit><latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit><latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit>

⌃(ek + eG,!) = ⌃(ek,!)<latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit><latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit><latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit><latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit>

















G(ek, eG,!)<latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit><latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit><latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit><latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit>


(5) periodic CPT, contd.


• apply CPT to using ref.sys.!

• this is diagonalized by VW:!

• implicit periodization, identifying!

• self-energy is discontinuous:

H<latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit><latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit><latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit><latexit sha1_base64="FNC4CL/LxFMEdg0V4Op3D7z3YgQ=">AAAB83icbVBNSwMxFHxbv2r9qnr0EiyCp7JbBD0WvPRYwW0L3aVk02wbmk2WJCuUpX/DiwdFvPpnvPlvzLZ70NaBwDDzhvcyUcqZNq777VS2tnd296r7tYPDo+OT+ulZT8tMEeoTyaUaRFhTzgT1DTOcDlJFcRJx2o9m94Xff6JKMykezTylYYIngsWMYGOlIJDWLLJ5ZzGqN9ymuwTaJF5JGlCiO6p/BWNJsoQKQzjWeui5qQlzrAwjnC5qQaZpiskMT+jQUoETqsN8efMCXVlljGKp7BMGLdXfiRwnWs+TyE4m2Ez1uleI/3nDzMR3Yc5EmhkqyGpRnHFkJCoKQGOmKDF8bgkmitlbEZlihYmxNdVsCd76lzdJr9X03Kb3cNNot8o6qnABl3ANHtxCGzrQBR8IpPAMr/DmZM6L8+58rEYrTpk5hz9wPn8AcbqR4g==</latexit> H 0

<latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit><latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit><latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit><latexit sha1_base64="J9YXspHhLARqn17nUb3ZAwoQgD8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mKoMeClx4r2g9oQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJXeZ3nlBpHstHM03Qj+hI8pAzaqz00LgsDcoVt+ouQNaJl5MK5GgOyl/9YczSCKVhgmrd89zE+DOqDGcC56V+qjGhbEJH2LNU0gi1P1ucOicXVhmSMFa2pCEL9ffEjEZaT6PAdkbUjPWql4n/eb3UhLf+jMskNSjZclGYCmJikv1NhlwhM2JqCWWK21sJG1NFmbHpZCF4qy+vk3at6rlV7/66Uq/lcRThDM7hCjy4gTo0oAktYDCCZ3iFN0c4L86787FsLTj5zCn8gfP5Ay19jQE=</latexit>




G(!) =1

! + µ � t + ⌃(!). (48)




G(

ek +

eG,!) =

1

! + µ � "(ek +

eG) +⌃(

eG,!)

. (49)






ek +


ek +

eG,!) ⌘ G(

ek, e

G,!)




ek +

eG) = ⌃(

eG) is e





⌃RR

0�(!) = �RR

0UhnR��i + �

RR

0U2hn

R��i(1 � hnR��i)

! + µ � U(1 � hnR��i)

, (50)




G(!) =1

! + µ � t + ⌃(!). (48)




G(

ek +

eG,!) =

1

! + µ � "(ek +

eG) +⌃(

eG,!)

. (49)






ek +


ek +

eG,!) ⌘ G(

ek, e

G,!)




ek +

eG) = ⌃(

eG) is e





⌃RR

0�(!) = �RR

0UhnR��i + �

RR

0U2hn

R��i(1 � hnR��i)

! + µ � U(1 � hnR��i)

, (50)

G(ek, eG,!) ⌘ G(k,!)<latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit><latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit><latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit><latexit sha1_base64="b07UeLJs//C+4s3nW8vETbMQqwE=">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</latexit>

⌃(ek + eG,!) = ⌃(ek,!)<latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit><latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit><latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit><latexit sha1_base64="t74jgGmy+B+WnHQ7HjiGVY/QDPU=">AAACWnichVFdS8MwFE3r5zo/5sebL8EhKMpoRdAXQfBBHyc6HaxjpOndDEuamqTKKPuTvojgXxFMtyHqFC+EHM45N7k5iVLOtPH9V8edmZ2bX1gseeWl5ZXVytr6rZaZotCgkkvVjIgGzhJoGGY4NFMFREQc7qL+eaHfPYLSTCY3ZpBCW5BewrqMEmOpTuUhvGY9QXbDJxaDYTyGPIwkj/VA2A33h3gf/6FdDA9CKaBH9rxT779jPq1ep1L1a/6o8DQIJqCKJlXvVJ7DWNJMQGIoJ1q3Aj817ZwowyiHoRdmGlJC+6QHLQsTIkC381E0Q7xjmRh3pbIrMXjEfu3IidDFkNYpiLnXP7WC/E1rZaZ70s5ZkmYGEjq+qJtxbCQucsYxU0ANH1hAqGJ2VkzviSLU2N8oQgh+Pnka3B7WAr8WXB1Vzw4ncSyiLbSNdlGAjtEZukR11EAUvaB3Z95ZcN5c1y255bHVdSY9G+hbuZsfmuS0nA==</latexit>




















reciprocal space





t’: disconnected clusters with periodic b.c. !modify the parameters of H rather than H’:!

G(ek, eG,!)<latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit><latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit><latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit><latexit sha1_base64="qQ3DNwjSaOKiYbh5IecloEsRtuU=">AAACJnicdVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6KIuK9gHtKVkMrdtaGYyJBmlDP0aN/6KGxcVEXd+ipl2FtrqgZDDOfdy7z1exJnSjvNpLS2vrK6t5zbym1vbO7v23n5diVhSqFHBhWx6RAFnIdQ00xyakQQSeBwa3vAm9RsPIBUT4b0eRdAJSD9kPUaJNlLXvqoU24/MB824D0nbE9xXo8B8eDg+/cepGEcE0CcnXbvglJwp8CJxM1JAGapde9L2BY0DCDXlRKmW60S6kxCpGeUwzrdjBRGhQ9KHlqEhCUB1kumZY3xsFB/3hDQv1Hiq/uxISKDSFU1lQPRAzXup+JfXinXvspOwMIo1hHQ2qBdzrAVOM8M+k0A1HxlCqGRmV0wHRBKqTbJ5E4I7f/IiqZ+VXKfk3p0XytdZHDl0iI5QEbnoApXRLaqiGqLoCb2gCXqznq1X6936mJUuWVnPAfoF6+sb3CqmmA==</latexit>

⌃( eG,!)<latexit sha1_base64="fDxJJ/LbKEIf4K7QGDlIZqmxzug=">AAACEXicbVDLSgMxFM34rPVVdekmWIQKUmZE0GXRhS4r2gd0SslkbtvQZDIkGaUM/QU3/oobF4q4defOvzHTdqGtB0IO59zLvfcEMWfauO63s7C4tLyymlvLr29sbm0XdnbrWiaKQo1KLlUzIBo4i6BmmOHQjBUQEXBoBIPLzG/cg9JMRndmGENbkF7EuowSY6VOoeTfsp4gJf+BhWAYDyH1A8lDPRT2w1ejY18K6JGjTqHolt0x8DzxpqSIpqh2Cl9+KGkiIDKUE61bnhubdkqUYZTDKO8nGmJCB6QHLUsjIkC30/FFI3xolRB3pbIvMnis/u5IidDZirZSENPXs14m/ue1EtM9b6csihMDEZ0M6iYcG4mzeHDIFFDDh5YQqpjdFdM+UYQaG2LehuDNnjxP6idlzy17N6fFysU0jhzaRweohDx0hiroGlVRDVH0iJ7RK3pznpwX5935mJQuONOePfQHzucPkDedcw==</latexit><latexit sha1_base64="fDxJJ/LbKEIf4K7QGDlIZqmxzug=">AAACEXicbVDLSgMxFM34rPVVdekmWIQKUmZE0GXRhS4r2gd0SslkbtvQZDIkGaUM/QU3/oobF4q4defOvzHTdqGtB0IO59zLvfcEMWfauO63s7C4tLyymlvLr29sbm0XdnbrWiaKQo1KLlUzIBo4i6BmmOHQjBUQEXBoBIPLzG/cg9JMRndmGENbkF7EuowSY6VOoeTfsp4gJf+BhWAYDyH1A8lDPRT2w1ejY18K6JGjTqHolt0x8DzxpqSIpqh2Cl9+KGkiIDKUE61bnhubdkqUYZTDKO8nGmJCB6QHLUsjIkC30/FFI3xolRB3pbIvMnis/u5IidDZirZSENPXs14m/ue1EtM9b6csihMDEZ0M6iYcG4mzeHDIFFDDh5YQqpjdFdM+UYQaG2LehuDNnjxP6idlzy17N6fFysU0jhzaRweohDx0hiroGlVRDVH0iJ7RK3pznpwX5935mJQuONOePfQHzucPkDedcw==</latexit><latexit sha1_base64="fDxJJ/LbKEIf4K7QGDlIZqmxzug=">AAACEXicbVDLSgMxFM34rPVVdekmWIQKUmZE0GXRhS4r2gd0SslkbtvQZDIkGaUM/QU3/oobF4q4defOvzHTdqGtB0IO59zLvfcEMWfauO63s7C4tLyymlvLr29sbm0XdnbrWiaKQo1KLlUzIBo4i6BmmOHQjBUQEXBoBIPLzG/cg9JMRndmGENbkF7EuowSY6VOoeTfsp4gJf+BhWAYDyH1A8lDPRT2w1ejY18K6JGjTqHolt0x8DzxpqSIpqh2Cl9+KGkiIDKUE61bnhubdkqUYZTDKO8nGmJCB6QHLUsjIkC30/FFI3xolRB3pbIvMnis/u5IidDZirZSENPXs14m/ue1EtM9b6csihMDEZ0M6iYcG4mzeHDIFFDDh5YQqpjdFdM+UYQaG2LehuDNnjxP6idlzy17N6fFysU0jhzaRweohDx0hiroGlVRDVH0iJ7RK3pznpwX5935mJQuONOePfQHzucPkDedcw==</latexit><latexit sha1_base64="fDxJJ/LbKEIf4K7QGDlIZqmxzug=">AAACEXicbVDLSgMxFM34rPVVdekmWIQKUmZE0GXRhS4r2gd0SslkbtvQZDIkGaUM/QU3/oobF4q4defOvzHTdqGtB0IO59zLvfcEMWfauO63s7C4tLyymlvLr29sbm0XdnbrWiaKQo1KLlUzIBo4i6BmmOHQjBUQEXBoBIPLzG/cg9JMRndmGENbkF7EuowSY6VOoeTfsp4gJf+BhWAYDyH1A8lDPRT2w1ejY18K6JGjTqHolt0x8DzxpqSIpqh2Cl9+KGkiIDKUE61bnhubdkqUYZTDKO8nGmJCB6QHLUsjIkC30/FFI3xolRB3pbIvMnis/u5IidDZirZSENPXs14m/ue1EtM9b6csihMDEZ0M6iYcG4mzeHDIFFDDh5YQqpjdFdM+UYQaG2LehuDNnjxP6idlzy17N6fFysU0jhzaRweohDx0hiroGlVRDVH0iJ7RK3pznpwX5935mJQuONOePfQHzucPkDedcw==</latexit>

SELF-CONSISTENT CLUSTER EMBEDDING

cluster extensions of DMFT

impurity / cluster approach!

Hubbard-I approximation!CPT!

periodic CPT!

with self-consistent embedding!

DMFT!cellular DMFT!

DCA!

lattice model single-site mean-field theory

cluster mean-field theory

recall DMFTsingle-site

mean-field theorylattice model


impurity / cluster approaches with self-consistent embeddingHubbard-I approach [18] DMFT [1, 4]

CPT [24, 25] cellular DMFT (C-DMFT) [9, 8]simplified PC-DMFT periodized C-DMFT [32]

periodic CPT [30] DCA [7]

Table 2: Different cluster approximations. See text for discussion.

as an approximation to compute the Green’s function for the Hubbard model is the essence ofthe Hubbard-I approximation. (Let us note that, in a strict sense, the Hubbard-I approximationdoes include some self-consistency and thus does not perfectly fit to the classification, see Ref.[18]). The DMFT is a single-site mean-field theory as well (Lc = 1) but with a self-consistentembedding of the site. It is important to realize that this is a dramatic improvement. Even thoughthe DMFT self-energy is local, ⌃

RR

0�(!) = �RR

0⌃R

(!), it has a realistic ! dependence withthe typical branch cuts on the real axis rather than the simplistic single-pole structure of theHubbard-I self-energy. The self-consistent DMFT embedding scheme is constructed such thatone even recovers the exact solution of the Hubbard model in the limit, where the self-energy islocal in fact [31], namely in the limit of infinite spatial dimensions [1].

Let us briefly recall the main idea for the Hubbard model as the original lattice model. Toconstruct the single-site DMFT, we assume that the self-energy be local. In addition, a homo-geneous (and nonmagnetic) phase is anticipated such that ⌃

RR

0�(!) = �RR

0⌃(!). We note inpassing that this condition can be relaxed; this leads to real-space DMFT [33]. The most gen-eral reference system with a local self-energy and arbitrary !-dependence is given by a singleinteracting (“impurity”) site with U 6= 0 hybridizing with a continuum of noninteracting bathdegrees of freedom, i.e., by the single-impurity Anderson model

H 0=

X

�

"impc†�c� +

U

2

X

�

nimp,�nimp,�� +

X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)

The local Green’s function on the impurity site is

G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)

The bath parameters, namely the hybridization strengths Vk and the on-site energies "k, enterthe formalism via the hybridization function only:

�(!) =X

k

V 2k

! + µ � "k

. (53)

We will use the self-energy of the reference system as an approximation for the lattice model,⌃(!) = ⌃ 0

(!). Obviously, a condition is needed to fix the hybridization function and therewiththe parameters of the reference system. This self-consistency condition is obtained from theobservation that the skeleton-diagram expansion leads to a functional relation between the self-energy and the Green’s function, ⌃ =

b⌃[G], that is independent from the relation provided by

local self-energy generated by

local Green’s function on the impurity site:







RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)




hybridization function:







RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)




use SIAM as reference system:







RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)




derive self-consistency condition to fix the parameters of H’ !

MFT

recall DMFT







RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)












RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)











RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)











RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)





DMFT

recall DMFT







RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)












RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)











RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)











RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)





DMFT

self-consistency condition

skeleton-diagram expansion:

functional relation for the Hubbard model (infinite D)

diagrams: summary

Luttinger-Ward functional:

to be discussed:!• variational principle ?!• grand potential: proof ?!• tr ln (…) term is ill-defined !

= +

= + + +Y−

XIV Training Course in the Physics of Strongly Correlated Systems Salerno, October 2010———————————————————————————————————————————————————————————————˛

˛

˛

˛

˛

˛

˛

˛

˛

˛

Luttinger-Ward functional

define (with an additional factor (−T )):

= + + +ΦLuttinger, Ward (1960)

note: Φ = −T ln Z

what is Φ good for ?

Σ =1

T

δΦ

δGIMPORTANT !!!

Φ is like a potential for the self-energy !——————————————————————————————————————–proof:

note: Φ = bΦU[G]

– Φ is a functional of the Green’s function– the functional dependence is fixed by U (and independent of t)– bΦU[G] is a universal functional

– p. 163

self-energy, skeleton-diagram expansion:

Dyson’s equation

grand potential and LW functional:

derivative of the LW funct.:

⌦ = �+ tr lnG� tr(⌃G)

��[G]

�Gij�(i!n)= ⌃ji�(i!n)[G]

functional relation for the impurity Anderson model⌃ = ⌃[G(loc)]

<latexit sha1_base64="xkV+XxX8Lqm+vk2/+BB/WYsQPpU=">AAACMnicbVDLSgMxFM3UVx1fVZdugkWomzJTBN0IBRfqrqJ9QGcsmTRtQ5PJkGSEMsw3ufFLBBe6UMStH2Gm7cI+DoQczrmXe+8JIkaVdpx3K7eyura+kd+0t7Z3dvcK+wcNJWKJSR0LJmQrQIowGpK6ppqRViQJ4gEjzWB4lfnNJyIVFeGDHkXE56gf0h7FSBupU7j1AsG6asTNl3j3tM9RCi+hvURuz4jX6WPiSQ5LTODT1PbtTqHolJ0x4CJxp6QIpqh1Cq9eV+CYk1BjhpRqu06k/QRJTTEjqe3FikQID1GftA0NESfKT8Ynp/DEKF3YE9K8UMOx+r8jQVxlm5pKjvRAzXuZuMxrx7p34Sc0jGJNQjwZ1IsZ1AJm+cEulQRrNjIEYUnNrhAPkERYm5SzENz5kxdJo1J2nbJ7d1asVqZx5MEROAYl4IJzUAU3oAbqAINn8AY+wZf1Yn1Y39bPpDRnTXsOwQys3z8s5aqU</latexit><latexit sha1_base64="xkV+XxX8Lqm+vk2/+BB/WYsQPpU=">AAACMnicbVDLSgMxFM3UVx1fVZdugkWomzJTBN0IBRfqrqJ9QGcsmTRtQ5PJkGSEMsw3ufFLBBe6UMStH2Gm7cI+DoQczrmXe+8JIkaVdpx3K7eyura+kd+0t7Z3dvcK+wcNJWKJSR0LJmQrQIowGpK6ppqRViQJ4gEjzWB4lfnNJyIVFeGDHkXE56gf0h7FSBupU7j1AsG6asTNl3j3tM9RCi+hvURuz4jX6WPiSQ5LTODT1PbtTqHolJ0x4CJxp6QIpqh1Cq9eV+CYk1BjhpRqu06k/QRJTTEjqe3FikQID1GftA0NESfKT8Ynp/DEKF3YE9K8UMOx+r8jQVxlm5pKjvRAzXuZuMxrx7p34Sc0jGJNQjwZ1IsZ1AJm+cEulQRrNjIEYUnNrhAPkERYm5SzENz5kxdJo1J2nbJ7d1asVqZx5MEROAYl4IJzUAU3oAbqAINn8AY+wZf1Yn1Y39bPpDRnTXsOwQys3z8s5aqU</latexit><latexit sha1_base64="xkV+XxX8Lqm+vk2/+BB/WYsQPpU=">AAACMnicbVDLSgMxFM3UVx1fVZdugkWomzJTBN0IBRfqrqJ9QGcsmTRtQ5PJkGSEMsw3ufFLBBe6UMStH2Gm7cI+DoQczrmXe+8JIkaVdpx3K7eyura+kd+0t7Z3dvcK+wcNJWKJSR0LJmQrQIowGpK6ppqRViQJ4gEjzWB4lfnNJyIVFeGDHkXE56gf0h7FSBupU7j1AsG6asTNl3j3tM9RCi+hvURuz4jX6WPiSQ5LTODT1PbtTqHolJ0x4CJxp6QIpqh1Cq9eV+CYk1BjhpRqu06k/QRJTTEjqe3FikQID1GftA0NESfKT8Ynp/DEKF3YE9K8UMOx+r8jQVxlm5pKjvRAzXuZuMxrx7p34Sc0jGJNQjwZ1IsZ1AJm+cEulQRrNjIEYUnNrhAPkERYm5SzENz5kxdJo1J2nbJ7d1asVqZx5MEROAYl4IJzUAU3oAbqAINn8AY+wZf1Yn1Y39bPpDRnTXsOwQys3z8s5aqU</latexit><latexit sha1_base64="xkV+XxX8Lqm+vk2/+BB/WYsQPpU=">AAACMnicbVDLSgMxFM3UVx1fVZdugkWomzJTBN0IBRfqrqJ9QGcsmTRtQ5PJkGSEMsw3ufFLBBe6UMStH2Gm7cI+DoQczrmXe+8JIkaVdpx3K7eyura+kd+0t7Z3dvcK+wcNJWKJSR0LJmQrQIowGpK6ppqRViQJ4gEjzWB4lfnNJyIVFeGDHkXE56gf0h7FSBupU7j1AsG6asTNl3j3tM9RCi+hvURuz4jX6WPiSQ5LTODT1PbtTqHolJ0x4CJxp6QIpqh1Cq9eV+CYk1BjhpRqu06k/QRJTTEjqe3FikQID1GftA0NESfKT8Ynp/DEKF3YE9K8UMOx+r8jQVxlm5pKjvRAzXuZuMxrx7p34Sc0jGJNQjwZ1IsZ1AJm+cEulQRrNjIEYUnNrhAPkERYm5SzENz5kxdJo1J2nbJ7d1asVqZx5MEROAYl4IJzUAU3oAbqAINn8AY+wZf1Yn1Y39bPpDRnTXsOwQys3z8s5aqU</latexit>

⌃0 = ⌃[G(imp)]<latexit sha1_base64="Ci8WZnSTsmDG2/yw+IEE90vzlYk=">AAACM3icbVDLSsNAFJ34rPFVdelmsIh1U5Ii6EYouFBcVbQPSGKZTKbt0JkkzEyEEvJPbvwRF4K4UMSt/+Ck7cI+DgxzOOde7r3HjxmVyrLejaXlldW19cKGubm1vbNb3NtvyigRmDRwxCLR9pEkjIakoahipB0LgrjPSMsfXOV+64kISaPwQQ1j4nHUC2mXYqS01Cneun7EAjnk+kvde9rjKDuBl9BcoDtT4nX2mLqCwzLl8WlmemanWLIq1ghwntgTUgIT1DvFVzeIcMJJqDBDUjq2FSsvRUJRzEhmuokkMcID1COOpiHiRHrp6OYMHmslgN1I6BcqOFL/d6SIy3xTXcmR6stZLxcXeU6iuhdeSsM4USTE40HdhEEVwTxAGFBBsGJDTRAWVO8KcR8JhJWOOQ/Bnj15njSrFduq2HdnpVp1EkcBHIIjUAY2OAc1cAPqoAEweAZv4BN8GS/Gh/Ft/IxLl4xJzwGYgvH7B6nmqs0=</latexit><latexit sha1_base64="Ci8WZnSTsmDG2/yw+IEE90vzlYk=">AAACM3icbVDLSsNAFJ34rPFVdelmsIh1U5Ii6EYouFBcVbQPSGKZTKbt0JkkzEyEEvJPbvwRF4K4UMSt/+Ck7cI+DgxzOOde7r3HjxmVyrLejaXlldW19cKGubm1vbNb3NtvyigRmDRwxCLR9pEkjIakoahipB0LgrjPSMsfXOV+64kISaPwQQ1j4nHUC2mXYqS01Cneun7EAjnk+kvde9rjKDuBl9BcoDtT4nX2mLqCwzLl8WlmemanWLIq1ghwntgTUgIT1DvFVzeIcMJJqDBDUjq2FSsvRUJRzEhmuokkMcID1COOpiHiRHrp6OYMHmslgN1I6BcqOFL/d6SIy3xTXcmR6stZLxcXeU6iuhdeSsM4USTE40HdhEEVwTxAGFBBsGJDTRAWVO8KcR8JhJWOOQ/Bnj15njSrFduq2HdnpVp1EkcBHIIjUAY2OAc1cAPqoAEweAZv4BN8GS/Gh/Ft/IxLl4xJzwGYgvH7B6nmqs0=</latexit><latexit sha1_base64="Ci8WZnSTsmDG2/yw+IEE90vzlYk=">AAACM3icbVDLSsNAFJ34rPFVdelmsIh1U5Ii6EYouFBcVbQPSGKZTKbt0JkkzEyEEvJPbvwRF4K4UMSt/+Ck7cI+DgxzOOde7r3HjxmVyrLejaXlldW19cKGubm1vbNb3NtvyigRmDRwxCLR9pEkjIakoahipB0LgrjPSMsfXOV+64kISaPwQQ1j4nHUC2mXYqS01Cneun7EAjnk+kvde9rjKDuBl9BcoDtT4nX2mLqCwzLl8WlmemanWLIq1ghwntgTUgIT1DvFVzeIcMJJqDBDUjq2FSsvRUJRzEhmuokkMcID1COOpiHiRHrp6OYMHmslgN1I6BcqOFL/d6SIy3xTXcmR6stZLxcXeU6iuhdeSsM4USTE40HdhEEVwTxAGFBBsGJDTRAWVO8KcR8JhJWOOQ/Bnj15njSrFduq2HdnpVp1EkcBHIIjUAY2OAc1cAPqoAEweAZv4BN8GS/Gh/Ft/IxLl4xJzwGYgvH7B6nmqs0=</latexit><latexit sha1_base64="Ci8WZnSTsmDG2/yw+IEE90vzlYk=">AAACM3icbVDLSsNAFJ34rPFVdelmsIh1U5Ii6EYouFBcVbQPSGKZTKbt0JkkzEyEEvJPbvwRF4K4UMSt/+Ck7cI+DgxzOOde7r3HjxmVyrLejaXlldW19cKGubm1vbNb3NtvyigRmDRwxCLR9pEkjIakoahipB0LgrjPSMsfXOV+64kISaPwQQ1j4nHUC2mXYqS01Cneun7EAjnk+kvde9rjKDuBl9BcoDtT4nX2mLqCwzLl8WlmemanWLIq1ghwntgTUgIT1DvFVzeIcMJJqDBDUjq2FSsvRUJRzEhmuokkMcID1COOpiHiRHrp6OYMHmslgN1I6BcqOFL/d6SIy3xTXcmR6stZLxcXeU6iuhdeSsM4USTE40HdhEEVwTxAGFBBsGJDTRAWVO8KcR8JhJWOOQ/Bnj15njSrFduq2HdnpVp1EkcBHIIjUAY2OAc1cAPqoAEweAZv4BN8GS/Gh/Ft/IxLl4xJzwGYgvH7B6nmqs0=</latexit>

with this implies:⌃ = ⌃0<latexit sha1_base64="JTK0ewJ/KyaA+iuLMYBPnPnIxrg=">AAACFXicbVDLSgMxFM3UVx1foy7dBIvoQspMEXQjFNy4rGgf0BlKJpO2oZlkSDJCGfoTbvwVNy4UcSu482/MtLPQtgdCDufcy733hAmjSrvuj1VaWV1b3yhv2lvbO7t7zv5BS4lUYtLEggnZCZEijHLS1FQz0kkkQXHISDsc3eR++5FIRQV/0OOEBDEacNqnGGkj9ZxzPxQsUuPYfJl/TwcxmtjX9hL11LZ7TsWtulPAReIVpAIKNHrOtx8JnMaEa8yQUl3PTXSQIakpZmRi+6kiCcIjNCBdQzmKiQqy6VUTeGKUCPaFNI9rOFX/dmQoVvmKpjJGeqjmvVxc5nVT3b8KMsqTVBOOZ4P6KYNawDwiGFFJsGZjQxCW1OwK8RBJhLUJMg/Bmz95kbRqVc+tencXlXqtiKMMjsAxOAMeuAR1cAsaoAkweAIv4A28W8/Wq/Vhfc5KS1bRcwj+wfr6Beygno8=</latexit><latexit sha1_base64="JTK0ewJ/KyaA+iuLMYBPnPnIxrg=">AAACFXicbVDLSgMxFM3UVx1foy7dBIvoQspMEXQjFNy4rGgf0BlKJpO2oZlkSDJCGfoTbvwVNy4UcSu482/MtLPQtgdCDufcy733hAmjSrvuj1VaWV1b3yhv2lvbO7t7zv5BS4lUYtLEggnZCZEijHLS1FQz0kkkQXHISDsc3eR++5FIRQV/0OOEBDEacNqnGGkj9ZxzPxQsUuPYfJl/TwcxmtjX9hL11LZ7TsWtulPAReIVpAIKNHrOtx8JnMaEa8yQUl3PTXSQIakpZmRi+6kiCcIjNCBdQzmKiQqy6VUTeGKUCPaFNI9rOFX/dmQoVvmKpjJGeqjmvVxc5nVT3b8KMsqTVBOOZ4P6KYNawDwiGFFJsGZjQxCW1OwK8RBJhLUJMg/Bmz95kbRqVc+tencXlXqtiKMMjsAxOAMeuAR1cAsaoAkweAIv4A28W8/Wq/Vhfc5KS1bRcwj+wfr6Beygno8=</latexit><latexit sha1_base64="JTK0ewJ/KyaA+iuLMYBPnPnIxrg=">AAACFXicbVDLSgMxFM3UVx1foy7dBIvoQspMEXQjFNy4rGgf0BlKJpO2oZlkSDJCGfoTbvwVNy4UcSu482/MtLPQtgdCDufcy733hAmjSrvuj1VaWV1b3yhv2lvbO7t7zv5BS4lUYtLEggnZCZEijHLS1FQz0kkkQXHISDsc3eR++5FIRQV/0OOEBDEacNqnGGkj9ZxzPxQsUuPYfJl/TwcxmtjX9hL11LZ7TsWtulPAReIVpAIKNHrOtx8JnMaEa8yQUl3PTXSQIakpZmRi+6kiCcIjNCBdQzmKiQqy6VUTeGKUCPaFNI9rOFX/dmQoVvmKpjJGeqjmvVxc5nVT3b8KMsqTVBOOZ4P6KYNawDwiGFFJsGZjQxCW1OwK8RBJhLUJMg/Bmz95kbRqVc+tencXlXqtiKMMjsAxOAMeuAR1cAsaoAkweAIv4A28W8/Wq/Vhfc5KS1bRcwj+wfr6Beygno8=</latexit><latexit sha1_base64="JTK0ewJ/KyaA+iuLMYBPnPnIxrg=">AAACFXicbVDLSgMxFM3UVx1foy7dBIvoQspMEXQjFNy4rGgf0BlKJpO2oZlkSDJCGfoTbvwVNy4UcSu482/MtLPQtgdCDufcy733hAmjSrvuj1VaWV1b3yhv2lvbO7t7zv5BS4lUYtLEggnZCZEijHLS1FQz0kkkQXHISDsc3eR++5FIRQV/0OOEBDEacNqnGGkj9ZxzPxQsUuPYfJl/TwcxmtjX9hL11LZ7TsWtulPAReIVpAIKNHrOtx8JnMaEa8yQUl3PTXSQIakpZmRi+6kiCcIjNCBdQzmKiQqy6VUTeGKUCPaFNI9rOFX/dmQoVvmKpjJGeqjmvVxc5nVT3b8KMsqTVBOOZ4P6KYNawDwiGFFJsGZjQxCW1OwK8RBJhLUJMg/Bmz95kbRqVc+tencXlXqtiKMMjsAxOAMeuAR1cAsaoAkweAIv4A28W8/Wq/Vhfc5KS1bRcwj+wfr6Beygno8=</latexit>

G(imp)(!) = G(loc)(!)<latexit sha1_base64="OjUgbk2H3GYs0dXHFiE8E6vcHIA=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EQoudFnBXqAzlkyaaUOTyZBkhDL0Kdz4Km5cKOJW3Pk2pu2I2vpD4Oc753By/iBmVGnH+bRyS8srq2v59cLG5tb2jr2711QikZg0sGBCtgOkCKMRaWiqGWnHkiAeMNIKhheTeuuOSEVFdKNHMfE56kc0pBhpg7r28eVt6kkOS5TH5XHJE5z0URmew2/OBP7hXbvoVJyp4KJxM1MEmepd+8PrCZxwEmnMkFId14m1nyKpKWZkXPASRWKEh6hPOsZGiBPlp9OzxvDIkB4MhTQv0nBKf0+kiCs14oHp5EgP1HxtAv+rdRIdnvkpjeJEkwjPFoUJg1rASUawRyXBmo2MQVhS81eIB0girE2SBROCO3/yomlWK65Tca9PirVqFkceHIBDUAIuOAU1cAXqoAEwuAeP4Bm8WA/Wk/Vqvc1ac1Y2sw/+yHr/Ag13nVw=</latexit><latexit sha1_base64="OjUgbk2H3GYs0dXHFiE8E6vcHIA=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EQoudFnBXqAzlkyaaUOTyZBkhDL0Kdz4Km5cKOJW3Pk2pu2I2vpD4Oc753By/iBmVGnH+bRyS8srq2v59cLG5tb2jr2711QikZg0sGBCtgOkCKMRaWiqGWnHkiAeMNIKhheTeuuOSEVFdKNHMfE56kc0pBhpg7r28eVt6kkOS5TH5XHJE5z0URmew2/OBP7hXbvoVJyp4KJxM1MEmepd+8PrCZxwEmnMkFId14m1nyKpKWZkXPASRWKEh6hPOsZGiBPlp9OzxvDIkB4MhTQv0nBKf0+kiCs14oHp5EgP1HxtAv+rdRIdnvkpjeJEkwjPFoUJg1rASUawRyXBmo2MQVhS81eIB0girE2SBROCO3/yomlWK65Tca9PirVqFkceHIBDUAIuOAU1cAXqoAEwuAeP4Bm8WA/Wk/Vqvc1ac1Y2sw/+yHr/Ag13nVw=</latexit><latexit sha1_base64="OjUgbk2H3GYs0dXHFiE8E6vcHIA=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EQoudFnBXqAzlkyaaUOTyZBkhDL0Kdz4Km5cKOJW3Pk2pu2I2vpD4Oc753By/iBmVGnH+bRyS8srq2v59cLG5tb2jr2711QikZg0sGBCtgOkCKMRaWiqGWnHkiAeMNIKhheTeuuOSEVFdKNHMfE56kc0pBhpg7r28eVt6kkOS5TH5XHJE5z0URmew2/OBP7hXbvoVJyp4KJxM1MEmepd+8PrCZxwEmnMkFId14m1nyKpKWZkXPASRWKEh6hPOsZGiBPlp9OzxvDIkB4MhTQv0nBKf0+kiCs14oHp5EgP1HxtAv+rdRIdnvkpjeJEkwjPFoUJg1rASUawRyXBmo2MQVhS81eIB0girE2SBROCO3/yomlWK65Tca9PirVqFkceHIBDUAIuOAU1cAXqoAEwuAeP4Bm8WA/Wk/Vqvc1ac1Y2sw/+yHr/Ag13nVw=</latexit><latexit sha1_base64="OjUgbk2H3GYs0dXHFiE8E6vcHIA=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EQoudFnBXqAzlkyaaUOTyZBkhDL0Kdz4Km5cKOJW3Pk2pu2I2vpD4Oc753By/iBmVGnH+bRyS8srq2v59cLG5tb2jr2711QikZg0sGBCtgOkCKMRaWiqGWnHkiAeMNIKhheTeuuOSEVFdKNHMfE56kc0pBhpg7r28eVt6kkOS5TH5XHJE5z0URmew2/OBP7hXbvoVJyp4KJxM1MEmepd+8PrCZxwEmnMkFId14m1nyKpKWZkXPASRWKEh6hPOsZGiBPlp9OzxvDIkB4MhTQv0nBKf0+kiCs14oHp5EgP1HxtAv+rdRIdnvkpjeJEkwjPFoUJg1rASUawRyXBmo2MQVhS81eIB0girE2SBROCO3/yomlWK65Tca9PirVqFkceHIBDUAIuOAU1cAXqoAEwuAeP4Bm8WA/Wk/Vqvc1ac1Y2sw/+yHr/Ag13nVw=</latexit>

DMFT

DMFT self-consistency condition







RR

0�(!) = �RR

0⌃R



RR

0�(!) = �RR


H 0=

X

�

"impc†�c� +

U

2

X

�


X

k�

"ka†k�ak� +

X

k�

(Vkc†�ak� + H.c.) . (51)


G(imp)(!) =

1

! + µ � "imp � �(!) � ⌃ 0(!)

. (52)


�(!) =X

k

V 2k

! + µ � "k

. (53)




where

solving for the hybridization function:

�(!) =X

k

V 2

k

! + µ� "k= ! + µ� "

imp

� ⌃(!)� 1

G(loc)(!)<latexit sha1_base64="yLZaclTqcwwXwaKSmHhQsEjPL4g=">AAAChXicbVFda9swFJW9j3beR9PtcS9iYZAyEuxQtr6MFTbYHju2pIUoNdfKdSosWUaSC8H4n+xX7W3/Zorjsa7dBcHhnHN1pXOzSgrr4vhXEN67/+Dh3v6j6PGTp88OBofP51bXhuOMa6nNRQYWpShx5oSTeFEZBJVJPM+Kj1v9/BqNFbr87jYVLhWsS5ELDs5T6eAH+4TSwYhphWs4otH7iNlapQVluQHezNPicto2O5m+YaqmY3YNBisrpL+gaLuWXqadTm8aGmYUFapqt/w3sVZ/Z437GUnbfL7sfCOp+VH7x9Cmg2E8ibuid0HSgyHp6ywd/GQrzWuFpeMSrF0kceWWDRgnuMQ2YrXFCngBa1x4WIJCu2y6FFv62jMrmmvjT+lox97saEBZu1GZdypwV/a2tiX/py1ql58sG1FWtcOS7wbltaRO0+1K6EoY5E5uPABuhH8r5Vfgk3F+cZEPIbn95btgPp0k8ST5ejw8nfZx7JOX5BUZkYS8I6fkCzkjM8KDMBgFSTAN98JxeBy+3VnDoO95Qf6p8MNvRD6/tg==</latexit><latexit sha1_base64="yLZaclTqcwwXwaKSmHhQsEjPL4g=">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</latexit><latexit sha1_base64="yLZaclTqcwwXwaKSmHhQsEjPL4g=">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</latexit><latexit sha1_base64="yLZaclTqcwwXwaKSmHhQsEjPL4g=">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</latexit>

DMFT self-consistency cycle

⌃(!)<latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit><latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit><latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit><latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit>

G(!)<latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit><latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit><latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit><latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit>

G(loc)(!)<latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit><latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit><latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit><latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit>

�(!)<latexit sha1_base64="xSQRRIe8eM7Y/gYpR5ito95LLnE=">AAAB+HicbVDLSsNAFJ3UV62PRl26CRahbkpSBF0WdOGygn1AE8pketMOnUeYmQg19EvcuFDErZ/izr8xabPQ1gMXDufcy733hDGj2rjut1Xa2Nza3invVvb2Dw6r9tFxV8tEEegQyaTqh1gDowI6hhoG/VgB5iGDXji9yf3eIyhNpXgwsxgCjseCRpRgk0lDu+rfAjO47ksOY3xRGdo1t+Eu4KwTryA1VKA9tL/8kSQJB2EIw1oPPDc2QYqVoYTBvOInGmJMpngMg4wKzEEH6eLwuXOeKSMnkiorYZyF+nsixVzrGQ+zTo7NRK96ufifN0hMdB2kVMSJAUGWi6KEOUY6eQrOiCoghs0ygomi2a0OmWCFicmyykPwVl9eJ91mw3Mb3v1lrdUs4iijU3SG6shDV6iF7lAbdRBBCXpGr+jNerJerHfrY9lasoqZE/QH1ucPbBqSNw==</latexit><latexit sha1_base64="xSQRRIe8eM7Y/gYpR5ito95LLnE=">AAAB+HicbVDLSsNAFJ3UV62PRl26CRahbkpSBF0WdOGygn1AE8pketMOnUeYmQg19EvcuFDErZ/izr8xabPQ1gMXDufcy733hDGj2rjut1Xa2Nza3invVvb2Dw6r9tFxV8tEEegQyaTqh1gDowI6hhoG/VgB5iGDXji9yf3eIyhNpXgwsxgCjseCRpRgk0lDu+rfAjO47ksOY3xRGdo1t+Eu4KwTryA1VKA9tL/8kSQJB2EIw1oPPDc2QYqVoYTBvOInGmJMpngMg4wKzEEH6eLwuXOeKSMnkiorYZyF+nsixVzrGQ+zTo7NRK96ufifN0hMdB2kVMSJAUGWi6KEOUY6eQrOiCoghs0ygomi2a0OmWCFicmyykPwVl9eJ91mw3Mb3v1lrdUs4iijU3SG6shDV6iF7lAbdRBBCXpGr+jNerJerHfrY9lasoqZE/QH1ucPbBqSNw==</latexit><latexit sha1_base64="xSQRRIe8eM7Y/gYpR5ito95LLnE=">AAAB+HicbVDLSsNAFJ3UV62PRl26CRahbkpSBF0WdOGygn1AE8pketMOnUeYmQg19EvcuFDErZ/izr8xabPQ1gMXDufcy733hDGj2rjut1Xa2Nza3invVvb2Dw6r9tFxV8tEEegQyaTqh1gDowI6hhoG/VgB5iGDXji9yf3eIyhNpXgwsxgCjseCRpRgk0lDu+rfAjO47ksOY3xRGdo1t+Eu4KwTryA1VKA9tL/8kSQJB2EIw1oPPDc2QYqVoYTBvOInGmJMpngMg4wKzEEH6eLwuXOeKSMnkiorYZyF+nsixVzrGQ+zTo7NRK96ufifN0hMdB2kVMSJAUGWi6KEOUY6eQrOiCoghs0ygomi2a0OmWCFicmyykPwVl9eJ91mw3Mb3v1lrdUs4iijU3SG6shDV6iF7lAbdRBBCXpGr+jNerJerHfrY9lasoqZE/QH1ucPbBqSNw==</latexit><latexit sha1_base64="xSQRRIe8eM7Y/gYpR5ito95LLnE=">AAAB+HicbVDLSsNAFJ3UV62PRl26CRahbkpSBF0WdOGygn1AE8pketMOnUeYmQg19EvcuFDErZ/izr8xabPQ1gMXDufcy733hDGj2rjut1Xa2Nza3invVvb2Dw6r9tFxV8tEEegQyaTqh1gDowI6hhoG/VgB5iGDXji9yf3eIyhNpXgwsxgCjseCRpRgk0lDu+rfAjO47ksOY3xRGdo1t+Eu4KwTryA1VKA9tL/8kSQJB2EIw1oPPDc2QYqVoYTBvOInGmJMpngMg4wKzEEH6eLwuXOeKSMnkiorYZyF+nsixVzrGQ+zTo7NRK96ufifN0hMdB2kVMSJAUGWi6KEOUY6eQrOiCoghs0ygomi2a0OmWCFicmyykPwVl9eJ91mw3Mb3v1lrdUs4iijU3SG6shDV6iF7lAbdRBBCXpGr+jNerJerHfrY9lasoqZE/QH1ucPbBqSNw==</latexit>

Anderson!impurity !model

solve use Dyson equation

pick local element

use DMFT s.c. conditiondefines

cellular DMFT

⌃(!)<latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit><latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit><latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit><latexit sha1_base64="u6DrSDYd9twBT0jMTNKI5Iu7wfc=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxC3ZSkCLosuHFZ0T6gCWUynaRD5xFmJkIN/RI3LhRx66e482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dV9+i4q2WmMOlgyaTqR0gTRgXpGGoY6aeKIB4x0osmN3O/90iUplI8mGlKQo4SQWOKkbHS0K0G9zThqB5IThJ0AYduzWt4C8B14hekBgq0h+5XMJI440QYzJDWA99LTZgjZShmZFYJMk1ShCcoIQNLBeJEh/ni8Bk8t8oIxlLZEgYu1N8TOeJaT3lkOzkyY73qzcX/vEFm4uswpyLNDBF4uSjOGDQSzlOAI6oINmxqCcKK2lshHiOFsLFZVWwI/urL66TbbPhew7+7rLWaRRxlcArOQB344Aq0wC1ogw7AIAPP4BW8OU/Oi/PufCxbS04xcwL+wPn8AZiEklQ=</latexit>

G(!)<latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit><latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit><latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit><latexit sha1_base64="nMU7ROHUp5VkCeGgd+uBfi4x9Xg=">AAAB/3icbVBLSwMxGMz6rPW1KnjxEixCvZTdIuix4EGPFewDukvJZtM2NI8lyQpl7cG/4sWDIl79G978N2bbPWjrQMgw831kMlHCqDae9+2srK6tb2yWtsrbO7t7++7BYVvLVGHSwpJJ1Y2QJowK0jLUMNJNFEE8YqQTja9zv/NAlKZS3JtJQkKOhoIOKEbGSn33OIgki/WE2yu7mVYDyckQnffdilfzZoDLxC9IBRRo9t2vIJY45UQYzJDWPd9LTJghZShmZFoOUk0ShMdoSHqWCsSJDrNZ/ik8s0oMB1LZIwycqb83MsR1HtFOcmRGetHLxf+8XmoGV2FGRZIaIvD8oUHKoJEwLwPGVBFs2MQShBW1WSEeIYWwsZWVbQn+4peXSbte872af3dRadSLOkrgBJyCKvDBJWiAW9AELYDBI3gGr+DNeXJenHfnYz664hQ7R+APnM8f+O2V/w==</latexit>

G(loc)(!)<latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit><latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit><latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit><latexit sha1_base64="98LJZijHbkYterIvLEag+uu3kds=">AAACAHicbVC7SgNBFJ2NrxhfqxYWNoNBSJqwGwQtAxZaRjAPyK5hdjJJhsxjmZkVwrKNv2JjoYitn2Hn3zhJttDEAxcO59zLvfdEMaPaeN63U1hb39jcKm6Xdnb39g/cw6O2lonCpIUlk6obIU0YFaRlqGGkGyuCeMRIJ5pcz/zOI1GaSnFvpjEJORoJOqQYGSv13ZP0JntIA8VhhUlczSqB5GSEqn237NW8OeAq8XNSBjmaffcrGEiccCIMZkjrnu/FJkyRMhQzkpWCRJMY4QkakZ6lAnGiw3T+QAbPrTKAQ6lsCQPn6u+JFHGtpzyynRyZsV72ZuJ/Xi8xw6swpSJODBF4sWiYMGgknKUBB1QRbNjUEoQVtbdCPEYKYWMzK9kQ/OWXV0m7XvO9mn93UW7U8ziK4BScgQrwwSVogFvQBC2AQQaewSt4c56cF+fd+Vi0Fpx85hj8gfP5A6CZlbY=</latexit>


Anderson!cluster !model


pick local element



diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛


define (with an additional factor (�T )):


note: � �= �T ln Z

what is � good for ?

� =1

T

��

�GIMPORTANT !!!

� is like a potential for the self-energy !——————————————————————————————————————–proof:

note: � = b�U[G]

– � is a functional of the Green’s function– the functional dependence is fixed by U (and independent of t)– b�U[G] is a universal functional

– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]

�Gij�(i�n)= �ji�(i�n)[G]

diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]


Fig. 10: Diagrammatic representation of Dyson’s equation, G(!) = G0(!) +

G0(!)⌃(!)G(!) (top). Representation of the skeleton-diagram expansion of the self-energy,⌃(!) = b

⌃[G](!) (bottom).

Dyson’s equation, see Fig. 10. On the single-site mean-field level or, equivalently, in the limit ofinfinite spatial dimensions, this functional is local, i.e., ⌃(!) = b⌃[G(loc)

](!) where G(loc)(!) =

GRR

(!) is the local (on-site) element of the Green’s function. Hence, the functional relationb⌃[..] is exactly the same as that of the reference system: ⌃ 0

(!) = b⌃[G(imp)](!). With ⌃(!) =

⌃ 0(!) this implies that

G(imp)(!) = G(loc)

(!) , (54)

which is the famous self-consistency condition of DMFT.

The step from single-site DMFT to cellular DMFT is conceptually simple and corresponds tothe step from the Hubbard-I approach to the CPT (see Tab. 2). One merely has to treat a finiteHubbard cluster with Lc > 1 sites as a “super site” and adopt the same strategy. The actual workto be done consists in the numerical computation of the self-energy ⌃ 0

rr

0(!) of the Anderson“cluster” model,

H 0=

X

rr

0�

trr

0c†r�cr0� +

U

2

X

r�

nr�nr�� +

X

k�

"rka

†rk�ark� +

X

rk�

(Vrkc

†r�ark� + H.c.) , (55)

which is a small Hubbard model of Lc correlated sites, where each correlated site r hybridizeswith an infinite number (k = 1, ...,1) of bath sites. The Lc ⇥ Lc self-energy matrix is takento approximate the “cluster local” self-energy of the Hubbard model, ⌃ e

R

eR,rr0(!) = ⌃ 0

rr

0(!).With this at hand, one can compute the “cluster-local” elements of the Green’s function of theoriginal model using Dyson’s equation, cf. Eq. (39),

G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0, (56)

and therewith the cluster Green’s function of the reference system via the C-DMFT self-consis-tency equation G(cluster)

rr

0 (!) = G(loc)rr

0 (!). Using the cluster analog of Eq. (52), where G(imp)(!)

is replaced by the Lc ⇥Lc matrix G

(cluster)(!), ⌃ 0

(!) by the matrix ⌃

0(!), "imp by the Lc ⇥Lc

intra-cluster hopping matrix t

0= t, and �(!) by the diagonal matrix �(!), the diagonal

elements of the latter are found as

�r

(!) =X

k�

V 2rk

! + µ � "rk

= ! + µ � trr

� ⌃rr

(!) � (G

(cluster))

�1rr

(!) . (57)

C-DMFT

Lichtenstein, !Katsnelson !(2000)!

Kotliar et al. !(2001)!

cellular DMFT




pick elements of G on the cluster


C-DMFT

⌃(!)<latexit sha1_base64="hPtGppEyAS1x3Jh7ejFxByMN/5I=">AAACBHicdVDLSgMxFM34rPVVdekmWIS6GTLTlra7ghuXFa0WOkPJpGkNJpMhyQhl6MKNv+LGhSJu/Qh3/o2ZWkFFD4QczrmXe++JEs60QejdWVhcWl5ZLawV1zc2t7ZLO7sXWqaK0C6RXKpehDXlLKZdwwynvURRLCJOL6Pr49y/vKFKMxmfm0lCQ4HHMRsxgo2VBqX9IJJ8qCfCfllwxsYCTyuBFHSMjwalMnJRHbWaCCLXr9b9Vs2SRhNVWz70XDRDGczRGZTegqEkqaCxIRxr3fdQYsIMK8MIp9NikGqaYHKNx7RvaYwF1WE2O2IKD60yhCOp7IsNnKnfOzIsdL6nrRTYXOnfXi7+5fVTM2qGGYuT1NCYfA4apRwaCfNE4JApSgyfWIKJYnZXSK6wwsTY3Io2hK9L4f/kwnc95HqntXLbn8dRAPvgAFSABxqgDU5AB3QBAbfgHjyCJ+fOeXCenZfP0gVn3rMHfsB5/QBjApiD</latexit><latexit sha1_base64="hPtGppEyAS1x3Jh7ejFxByMN/5I=">AAACBHicdVDLSgMxFM34rPVVdekmWIS6GTLTlra7ghuXFa0WOkPJpGkNJpMhyQhl6MKNv+LGhSJu/Qh3/o2ZWkFFD4QczrmXe++JEs60QejdWVhcWl5ZLawV1zc2t7ZLO7sXWqaK0C6RXKpehDXlLKZdwwynvURRLCJOL6Pr49y/vKFKMxmfm0lCQ4HHMRsxgo2VBqX9IJJ8qCfCfllwxsYCTyuBFHSMjwalMnJRHbWaCCLXr9b9Vs2SRhNVWz70XDRDGczRGZTegqEkqaCxIRxr3fdQYsIMK8MIp9NikGqaYHKNx7RvaYwF1WE2O2IKD60yhCOp7IsNnKnfOzIsdL6nrRTYXOnfXi7+5fVTM2qGGYuT1NCYfA4apRwaCfNE4JApSgyfWIKJYnZXSK6wwsTY3Io2hK9L4f/kwnc95HqntXLbn8dRAPvgAFSABxqgDU5AB3QBAbfgHjyCJ+fOeXCenZfP0gVn3rMHfsB5/QBjApiD</latexit><latexit sha1_base64="hPtGppEyAS1x3Jh7ejFxByMN/5I=">AAACBHicdVDLSgMxFM34rPVVdekmWIS6GTLTlra7ghuXFa0WOkPJpGkNJpMhyQhl6MKNv+LGhSJu/Qh3/o2ZWkFFD4QczrmXe++JEs60QejdWVhcWl5ZLawV1zc2t7ZLO7sXWqaK0C6RXKpehDXlLKZdwwynvURRLCJOL6Pr49y/vKFKMxmfm0lCQ4HHMRsxgo2VBqX9IJJ8qCfCfllwxsYCTyuBFHSMjwalMnJRHbWaCCLXr9b9Vs2SRhNVWz70XDRDGczRGZTegqEkqaCxIRxr3fdQYsIMK8MIp9NikGqaYHKNx7RvaYwF1WE2O2IKD60yhCOp7IsNnKnfOzIsdL6nrRTYXOnfXi7+5fVTM2qGGYuT1NCYfA4apRwaCfNE4JApSgyfWIKJYnZXSK6wwsTY3Io2hK9L4f/kwnc95HqntXLbn8dRAPvgAFSABxqgDU5AB3QBAbfgHjyCJ+fOeXCenZfP0gVn3rMHfsB5/QBjApiD</latexit><latexit sha1_base64="hPtGppEyAS1x3Jh7ejFxByMN/5I=">AAACBHicdVDLSgMxFM34rPVVdekmWIS6GTLTlra7ghuXFa0WOkPJpGkNJpMhyQhl6MKNv+LGhSJu/Qh3/o2ZWkFFD4QczrmXe++JEs60QejdWVhcWl5ZLawV1zc2t7ZLO7sXWqaK0C6RXKpehDXlLKZdwwynvURRLCJOL6Pr49y/vKFKMxmfm0lCQ4HHMRsxgo2VBqX9IJJ8qCfCfllwxsYCTyuBFHSMjwalMnJRHbWaCCLXr9b9Vs2SRhNVWz70XDRDGczRGZTegqEkqaCxIRxr3fdQYsIMK8MIp9NikGqaYHKNx7RvaYwF1WE2O2IKD60yhCOp7IsNnKnfOzIsdL6nrRTYXOnfXi7+5fVTM2qGGYuT1NCYfA4apRwaCfNE4JApSgyfWIKJYnZXSK6wwsTY3Io2hK9L4f/kwnc95HqntXLbn8dRAPvgAFSABxqgDU5AB3QBAbfgHjyCJ+fOeXCenZfP0gVn3rMHfsB5/QBjApiD</latexit>

G(loc)

rr0 (!)<latexit sha1_base64="q2J1rzVxFXhArE8YNp+I/7dZqhY=">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</latexit><latexit sha1_base64="q2J1rzVxFXhArE8YNp+I/7dZqhY=">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</latexit><latexit sha1_base64="q2J1rzVxFXhArE8YNp+I/7dZqhY=">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</latexit><latexit sha1_base64="q2J1rzVxFXhArE8YNp+I/7dZqhY=">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</latexit>


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]




⌃[G](!) (bottom).



GRR


(!) = b⌃[G(imp)](!). With ⌃(!) =


G(imp)(!) = G(loc)

(!) , (54)



rr


H 0=

X

rr

0�

trr

0c†r�cr0� +

U

2

X

r�

nr�nr�� +

X

k�

"rka

†rk�ark� +

X

rk�

(Vrkc

†r�ark� + H.c.) , (55)


R

eR,rr0(!) = ⌃ 0

rr


G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0, (56)


rr

0 (!) = G(loc)rr



(cluster)(!), ⌃ 0






�r

(!) =X

k�

V 2rk

! + µ � "rk

= ! + µ � trr

� ⌃rr

(!) � (G

(cluster))

�1rr

(!) . (57)


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]




⌃[G](!) (bottom).



GRR


(!) = b⌃[G(imp)](!). With ⌃(!) =


G(imp)(!) = G(loc)

(!) , (54)



rr


H 0=

X

rr

0�

trr

0c†r�cr0� +

U

2

X

r�

nr�nr�� +

X

k�

"rka

†rk�ark� +

X

rk�

(Vrkc

†r�ark� + H.c.) , (55)


R

eR,rr0(!) = ⌃ 0

rr


G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0, (56)


rr

0 (!) = G(loc)rr



(cluster)(!), ⌃ 0






�r

(!) =X

k�

V 2rk

! + µ � "rk

= ! + µ � trr

� ⌃rr

(!) � (G

(cluster))

�1rr

(!) . (57)

G(!)<latexit sha1_base64="giTiGmiiMFB/Scq99ITee1Y47e8=">AAAB/3icdVBLSwMxGMz6rPVVFbx4CRahXpbstqXtreBBjxXsA9pSsmm2Dc1uliQrlLUH/4oXD4p49W9489+YbSuo6EDIMPN9ZDJexJnSCH1YK6tr6xubma3s9s7u3n7u4LClRCwJbRLBhex4WFHOQtrUTHPaiSTFgcdp25tcpH77lkrFRHijpxHtB3gUMp8RrI00yB33PMGHahqYK7mcFXoioCN8PsjlkY3KqFZFENlusezWSoZUqqhYc6FjoznyYInGIPfeGwoSBzTUhGOlug6KdD/BUjPC6SzbixWNMJngEe0aGuKAqn4yzz+DZ0YZQl9Ic0IN5+r3jQQHKo1oJgOsx+q3l4p/ed1Y+9V+wsIo1jQki4f8mEMtYFoGHDJJieZTQzCRzGSFZIwlJtpUljUlfP0U/k9aru0g27ku5evuso4MOAGnoAAcUAF1cAUaoAkIuAMP4Ak8W/fWo/VivS5GV6zlzhH4AevtE2avlks=</latexit><latexit sha1_base64="giTiGmiiMFB/Scq99ITee1Y47e8=">AAAB/3icdVBLSwMxGMz6rPVVFbx4CRahXpbstqXtreBBjxXsA9pSsmm2Dc1uliQrlLUH/4oXD4p49W9489+YbSuo6EDIMPN9ZDJexJnSCH1YK6tr6xubma3s9s7u3n7u4LClRCwJbRLBhex4WFHOQtrUTHPaiSTFgcdp25tcpH77lkrFRHijpxHtB3gUMp8RrI00yB33PMGHahqYK7mcFXoioCN8PsjlkY3KqFZFENlusezWSoZUqqhYc6FjoznyYInGIPfeGwoSBzTUhGOlug6KdD/BUjPC6SzbixWNMJngEe0aGuKAqn4yzz+DZ0YZQl9Ic0IN5+r3jQQHKo1oJgOsx+q3l4p/ed1Y+9V+wsIo1jQki4f8mEMtYFoGHDJJieZTQzCRzGSFZIwlJtpUljUlfP0U/k9aru0g27ku5evuso4MOAGnoAAcUAF1cAUaoAkIuAMP4Ak8W/fWo/VivS5GV6zlzhH4AevtE2avlks=</latexit><latexit sha1_base64="giTiGmiiMFB/Scq99ITee1Y47e8=">AAAB/3icdVBLSwMxGMz6rPVVFbx4CRahXpbstqXtreBBjxXsA9pSsmm2Dc1uliQrlLUH/4oXD4p49W9489+YbSuo6EDIMPN9ZDJexJnSCH1YK6tr6xubma3s9s7u3n7u4LClRCwJbRLBhex4WFHOQtrUTHPaiSTFgcdp25tcpH77lkrFRHijpxHtB3gUMp8RrI00yB33PMGHahqYK7mcFXoioCN8PsjlkY3KqFZFENlusezWSoZUqqhYc6FjoznyYInGIPfeGwoSBzTUhGOlug6KdD/BUjPC6SzbixWNMJngEe0aGuKAqn4yzz+DZ0YZQl9Ic0IN5+r3jQQHKo1oJgOsx+q3l4p/ed1Y+9V+wsIo1jQki4f8mEMtYFoGHDJJieZTQzCRzGSFZIwlJtpUljUlfP0U/k9aru0g27ku5evuso4MOAGnoAAcUAF1cAUaoAkIuAMP4Ak8W/fWo/VivS5GV6zlzhH4AevtE2avlks=</latexit><latexit sha1_base64="giTiGmiiMFB/Scq99ITee1Y47e8=">AAAB/3icdVBLSwMxGMz6rPVVFbx4CRahXpbstqXtreBBjxXsA9pSsmm2Dc1uliQrlLUH/4oXD4p49W9489+YbSuo6EDIMPN9ZDJexJnSCH1YK6tr6xubma3s9s7u3n7u4LClRCwJbRLBhex4WFHOQtrUTHPaiSTFgcdp25tcpH77lkrFRHijpxHtB3gUMp8RrI00yB33PMGHahqYK7mcFXoioCN8PsjlkY3KqFZFENlusezWSoZUqqhYc6FjoznyYInGIPfeGwoSBzTUhGOlug6KdD/BUjPC6SzbixWNMJngEe0aGuKAqn4yzz+DZ0YZQl9Ic0IN5+r3jQQHKo1oJgOsx+q3l4p/ed1Y+9V+wsIo1jQki4f8mEMtYFoGHDJJieZTQzCRzGSFZIwlJtpUljUlfP0U/k9aru0g27ku5evuso4MOAGnoAAcUAF1cAUaoAkIuAMP4Ak8W/fWo/VivS5GV6zlzhH4AevtE2avlks=</latexit>

cellular DMFT





use C-DMFT s.c. conditiondefines

C-DMFT


G(loc)



diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]




⌃[G](!) (bottom).



GRR


(!) = b⌃[G(imp)](!). With ⌃(!) =


G(imp)(!) = G(loc)

(!) , (54)



rr


H 0=

X

rr

0�

trr

0c†r�cr0� +

U

2

X

r�

nr�nr�� +

X

k�

"rka

†rk�ark� +

X

rk�

(Vrkc

†r�ark� + H.c.) , (55)


R

eR,rr0(!) = ⌃ 0

rr


G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0, (56)


rr

0 (!) = G(loc)rr



(cluster)(!), ⌃ 0






�r

(!) =X

k�

V 2rk

! + µ � "rk

= ! + µ � trr

� ⌃rr

(!) � (G

(cluster))

�1rr

(!) . (57)


cellular DMFT





C-DMFT


G(loc)


�(!)<latexit sha1_base64="LbaMZNh4ZbO61lq72Pxho2uQbfU=">AAACBHicdVBNSwMxEM36WevXqkcvwSLoZclulba3gh48VrAqdEvJptM2NLtZkqxQlh68+Fe8eFDEqz/Cm//GtFZQ0Qchj/dmmJkXpYJrQ8i7Mze/sLi0XFgprq6tb2y6W9uXWmaKQZNJIdV1RDUInkDTcCPgOlVA40jAVTQ8mfhXN6A0l8mFGaXQjmk/4T3OqLFSx90NIym6ehTbLw9PQRg6PghlDH162HFLxCPHpFYlmHhB+TioHVlSqZJyLcC+R6YooRkaHfct7EqWxZAYJqjWLZ+kpp1TZTgTMC6GmYaUsiHtQ8vShMag2/n0iDHet0oX96SyLzF4qn7vyGmsJ3vaypiagf7tTcS/vFZmetV2zpM0M5Cwz0G9TGAj8SQR3OUKmBEjSyhT3O6K2YAqyozNrWhD+LoU/08uA88nnn9+VKoHszgKaBftoQPkowqqozPUQE3E0C26R4/oyblzHpxn5+WzdM6Z9eygH3BePwBX8Jh8</latexit><latexit sha1_base64="LbaMZNh4ZbO61lq72Pxho2uQbfU=">AAACBHicdVBNSwMxEM36WevXqkcvwSLoZclulba3gh48VrAqdEvJptM2NLtZkqxQlh68+Fe8eFDEqz/Cm//GtFZQ0Qchj/dmmJkXpYJrQ8i7Mze/sLi0XFgprq6tb2y6W9uXWmaKQZNJIdV1RDUInkDTcCPgOlVA40jAVTQ8mfhXN6A0l8mFGaXQjmk/4T3OqLFSx90NIym6ehTbLw9PQRg6PghlDH162HFLxCPHpFYlmHhB+TioHVlSqZJyLcC+R6YooRkaHfct7EqWxZAYJqjWLZ+kpp1TZTgTMC6GmYaUsiHtQ8vShMag2/n0iDHet0oX96SyLzF4qn7vyGmsJ3vaypiagf7tTcS/vFZmetV2zpM0M5Cwz0G9TGAj8SQR3OUKmBEjSyhT3O6K2YAqyozNrWhD+LoU/08uA88nnn9+VKoHszgKaBftoQPkowqqozPUQE3E0C26R4/oyblzHpxn5+WzdM6Z9eygH3BePwBX8Jh8</latexit><latexit sha1_base64="LbaMZNh4ZbO61lq72Pxho2uQbfU=">AAACBHicdVBNSwMxEM36WevXqkcvwSLoZclulba3gh48VrAqdEvJptM2NLtZkqxQlh68+Fe8eFDEqz/Cm//GtFZQ0Qchj/dmmJkXpYJrQ8i7Mze/sLi0XFgprq6tb2y6W9uXWmaKQZNJIdV1RDUInkDTcCPgOlVA40jAVTQ8mfhXN6A0l8mFGaXQjmk/4T3OqLFSx90NIym6ehTbLw9PQRg6PghlDH162HFLxCPHpFYlmHhB+TioHVlSqZJyLcC+R6YooRkaHfct7EqWxZAYJqjWLZ+kpp1TZTgTMC6GmYaUsiHtQ8vShMag2/n0iDHet0oX96SyLzF4qn7vyGmsJ3vaypiagf7tTcS/vFZmetV2zpM0M5Cwz0G9TGAj8SQR3OUKmBEjSyhT3O6K2YAqyozNrWhD+LoU/08uA88nnn9+VKoHszgKaBftoQPkowqqozPUQE3E0C26R4/oyblzHpxn5+WzdM6Z9eygH3BePwBX8Jh8</latexit><latexit sha1_base64="LbaMZNh4ZbO61lq72Pxho2uQbfU=">AAACBHicdVBNSwMxEM36WevXqkcvwSLoZclulba3gh48VrAqdEvJptM2NLtZkqxQlh68+Fe8eFDEqz/Cm//GtFZQ0Qchj/dmmJkXpYJrQ8i7Mze/sLi0XFgprq6tb2y6W9uXWmaKQZNJIdV1RDUInkDTcCPgOlVA40jAVTQ8mfhXN6A0l8mFGaXQjmk/4T3OqLFSx90NIym6ehTbLw9PQRg6PghlDH162HFLxCPHpFYlmHhB+TioHVlSqZJyLcC+R6YooRkaHfct7EqWxZAYJqjWLZ+kpp1TZTgTMC6GmYaUsiHtQ8vShMag2/n0iDHet0oX96SyLzF4qn7vyGmsJ3vaypiagf7tTcS/vFZmetV2zpM0M5Cwz0G9TGAj8SQR3OUKmBEjSyhT3O6K2YAqyozNrWhD+LoU/08uA88nnn9+VKoHszgKaBftoQPkowqqozPUQE3E0C26R4/oyblzHpxn5+WzdM6Z9eygH3BePwBX8Jh8</latexit>


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]


diagrams: summary



= +

= + + +Y−


˛

˛

˛

˛

˛

˛

˛

˛

˛






� =1

T

��

�GIMPORTANT !!!




– p. 163


Dyson’s equation



� = � + tr lnG � tr(�G)

��[G]




⌃[G](!) (bottom).



GRR


(!) = b⌃[G(imp)](!). With ⌃(!) =


G(imp)(!) = G(loc)

(!) , (54)



rr


H 0=

X

rr

0�

trr

0c†r�cr0� +

U

2

X

r�

nr�nr�� +

X

k�

"rka

†rk�ark� +

X

rk�

(Vrkc

†r�ark� + H.c.) , (55)


R

eR,rr0(!) = ⌃ 0

rr


G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0, (56)


rr

0 (!) = G(loc)rr



(cluster)(!), ⌃ 0






�r

(!) =X

k�

V 2rk

! + µ � "rk

= ! + µ � trr

� ⌃rr

(!) � (G

(cluster))

�1rr

(!) . (57)


periodized cellular DMFT





PC-DMFT


G(loc)




Biroli et al. !(2004)!.24 Michael Potthoff

The parameters of the reference system (55) are obtained as the poles and the weights of �r

(!),such that an updated cluster self-energy can be computed. This self-consistency cycle must beiterated until convergence is achieved.

The C-DMFT does not respect the translational symmetries of the underlying lattice. One wayto restore the correct symmetry is the so-called periodized C-DMFT (PC-DMFT) [32], see thirdrow in Tab. 2. Here, one employs the periodization operator bT to get a translationally invariantself-energy. The local Green’s function can thus be computed by U -Fourier transform andsummation of k. This yields the following PC-DMFT self-consistency equation:

G(cluster)rr

0 (!) = G(loc)rr

0 (!) =1

L

X

k

eik(r�r

0)

! + µ � "(k) � bT [⌃](k,!), (58)

The main difference as compared to the C-DMFT, Eq. (56), that the periodized self-energy isused at each step in the self-consistency cycle.

For the dynamical cluster approximation (DCA, fourth row in Tab. 2), one must replace theoriginal hopping t by t only [32]. Both, t and t, are invariant under superlattice translations. Sowe can compare t

rr

0(

ek) = (V

†tV )

rr

0(

ek) with t

rr

0(

ek) = (V

†tV )

rr

0(

ek). One easily finds that

these are equal up to a phase factor:

trr

0(

ek) =

1

Lc

X

eG

ei eG(r�r

0)"(ek +

eG)

=

Lc

L

X

eR

eR

0

e�iek( eR+r� eR

0�r

0)t eR+r, eR0+r

0

= e�iek(r�r

0)trr

0(

ek) . (59)

This is very easily implemented numerically. The DCA self-consistency condition reads:

G(cluster)rr

0 (!) = G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (60)

The decisive difference as compared to the C-DMFT is that t is also invariant under cycliccluster translations, as it is the case for t

0. Hence, all matrices are simultaneously diagonalizedby the cluster Fourier transformation W , and we get a scalar self-consistency equation:

G(cluster)(

eG,!) =

Lc

L

X

ek

1

! + µ � "(ek +

eG) � ⌃(

eG,!)

. (61)

Which of the three cluster approaches, the C-DMFT, the PC-DMFT, and the DCA is the bestone? This is a question that comes up immediately. An answer could be expected from a metatheory unifying all cluster approaches in a common theoretical framework. The self-energy-functional theory (SFT) [34,10,6] is in fact able to re-derive various approximations as station-ary points of a general functional ⌦

t,U [⌃] for a given original lattice model with parameters t

and U . A particular approximation is defined by the choice of a reference system, with possibly

dynamical cluster approximation

Anderson!cluster !

model with p.b.c.




PC-DMFT


G(loc)




Hettler et al. !(1998)!


Anderson!cluster !

model with p.b.c.

solve use Dyson equation with



DCA


G(loc)




Hettler et al. !(1998)! t 7! t

<latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit><latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit><latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit><latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit>





G(cluster)rr

0 (!) = G(loc)rr

0 (!) =1

L

X

k

eik(r�r

0)

! + µ � "(k) � bT [⌃](k,!), (58)



rr

0(

ek) = (V

†tV )

rr

0(

ek) with t

rr

0(

ek) = (V

†tV )

rr

0(



trr

0(

ek) =

1

Lc

X

eG

ei eG(r�r

0)"(ek +

eG)

=

Lc

L

X

eR

eR

0

e�iek( eR+r� eR

0�r

0)t eR+r, eR0+r

0

= e�iek(r�r

0)trr

0(

ek) . (59)


G(cluster)rr

0 (!) = G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (60)



G(cluster)(

eG,!) =

Lc

L

X

ek

1

! + µ � "(ek +

eG) � ⌃(

eG,!)

. (61)





Anderson!cluster !

model with p.b.c.

solve use Dyson equation with



DCA


G(loc)




Hettler et al. !(1998)! t 7! t

<latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit><latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit><latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">AAACGnicdVDLSgMxFM3UV62vUZdugkVwNWTq1Oqu4MZlBfuATimZNG1DM5MhyQhlmO9w46+4caGIO3Hj35hpK1jRCyEn5557c+8JYs6URujTKqysrq1vFDdLW9s7u3v2/kFLiUQS2iSCC9kJsKKcRbSpmea0E0uKw4DTdjC5yvPtOyoVE9Gtnsa0F+JRxIaMYG2ovu36geADNQ3NleoM+iGOlRbQF6Yqb5ouCzLYt8vIQTV05lYgcryqVzvPgXl7l1XoOmgWZbCIRt9+9weCJCGNNOFYqa6LYt1LsdSMcJqV/ETRGJMJHtGugREOqeqls9UyeGKYARwKaU6k4Yz9WZHiUOXDGWWI9Vj9zuXkX7luoocXvZRFcaJpROYfDRMOze65T3DAJCWaTw3ARDIzKyRjLDHRxs2SMeF7U/g/aFUcFznujVeuVxZ2FMEROAanwAU1UAfXoAGagIB78AiewYv1YD1Zr9bbXFqwFjWHYCmsjy8wQ6LK</latexit><latexit sha1_base64="QEtqcTZ6/2yCVEyUXVJoeFx+oL0=">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</latexit>





G(cluster)rr

0 (!) = G(loc)rr

0 (!) =1

L

X

k

eik(r�r

0)

! + µ � "(k) � bT [⌃](k,!), (58)



rr

0(

ek) = (V

†tV )

rr

0(

ek) with t

rr

0(

ek) = (V

†tV )

rr

0(



trr

0(

ek) =

1

Lc

X

eG

ei eG(r�r

0)"(ek +

eG)

=

Lc

L

X

eR

eR

0

e�iek( eR+r� eR

0�r

0)t eR+r, eR0+r

0

= e�iek(r�r

0)trr

0(

ek) . (59)


G(cluster)rr

0 (!) = G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (60)



G(cluster)(

eG,!) =

Lc

L

X

ek

1

! + µ � "(ek +

eG) � ⌃(

eG,!)

. (61)




DCA self-consistency equation:

DCA: k-space perspective





G(cluster)rr

0 (!) = G(loc)rr

0 (!) =1

L

X

k

eik(r�r

0)

! + µ � "(k) � bT [⌃](k,!), (58)



rr

0(

ek) = (V

†tV )

rr

0(

ek) with t

rr

0(

ek) = (V

†tV )

rr

0(



trr

0(

ek) =

1

Lc

X

eG

ei eG(r�r

0)"(ek +

eG)

=

Lc

L

X

eR

eR

0

e�iek( eR+r� eR

0�r

0)t eR+r, eR0+r

0

= e�iek(r�r

0)trr

0(

ek) . (59)


G(cluster)rr

0 (!) = G(loc)rr

0 (!) =Lc

L

X

ek2RSC

⇣1

! + µ � t(

ek) � ⌃(!)

⌘

rr

0. (60)



G(cluster)(

eG,!) =

Lc

L

X

ek

1

! + µ � "(ek +

eG) � ⌃(

eG,!)

. (61)




DCA self-consistency equation:




















reciprocal space





DMFT self-consistency equation:

G(imp)(!) = G(loc)(!) =1

L

X

k

1

! + µ� "(k)� ⌃(!)<latexit sha1_base64="/pcLO++FCYfbybtDIbVed3yMotw=">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</latexit><latexit sha1_base64="/pcLO++FCYfbybtDIbVed3yMotw=">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</latexit><latexit sha1_base64="/pcLO++FCYfbybtDIbVed3yMotw=">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</latexit><latexit sha1_base64="/pcLO++FCYfbybtDIbVed3yMotw=">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</latexit>

discard information on the !fine structure of the!momentum dependence !of the self-energy

keep overall, rough!information on the!

momentum dependence !

completely neglect the momentum dependence!of the self-energy

APPLICATIONS

Mott transition - DMFT

the numerically exact continuous time quantumMonte Carlo (CTQMC) method [13,14].

Results.—Fig. 1(a) shows the phase diagram of theHubbard model within cluster DMFT at half-filling in theabsence of long range order. For interaction strength U <Uc2ðTÞ, we find a metallic solution while forU >Uc1ðTÞ, aMott insulating solution exists. The two transition linesUc1ðTÞ and Uc2ðTÞ cross at a second order endpoint, attemperature TMIT # 0:09t and interaction strength UMIT #6:05t. It is clear that one of the most salient features of thesingle-site DMFT phase diagram [shown in Fig. 1(b)],namely, the existence of a first order phase transition,survives in plaquette DMFT.

Still there are substantial modifications to the single-siteDMFT results when U=t is close to its critical value.Namely, (i) Strong short ranged antiferromagnetic correla-tions significantly reduce the value of critical U at whichthe second order endpoint occurs. Note that the plaquette-DMFT critical Uð#6:05tÞ is in very favorable agreementwith the Monte Carlo crossover U at which the pseudogapdevelops at intermediate temperatures accessible by deter-minantal Monte Carlo calculations (Fig. 5 in Ref. [15]).This criticalU will increase if the system is more frustratedat short distance. For example, the inclusion of the nextnearest hopping t0 has this effect and was studied inRef. [16]. (ii) The shape of the coexistence region, whereboth metallic and insulating solutions exist, is significantlydifferent. The high temperature crossover lines (dashedline above T # 0:1t in Fig. 1) are similar since at hightemperature the entropy of the paramagnetic insulator is ofthe order of logð2Þ in both cluster and single-site approach.As the temperature is increased, the large entropy insulat-ing state wins over the lower entropy metallic state. At lowtemperature, the situation is very different. In single-siteDMFT, the metal wins at low temperature in the transitionregion because the emergence of the itinerant quasiparticleinside the Mott gap lowers the free energy of the stronglydisorderedMott state. In the cluster case, the Mott insulatorat very low temperature is very different and has smallentropy due to short range singlet formation. The smallentropy of this state can be confirmed by the ‘‘valencehistogram’’ shown in the inset of Fig. 1(a). The hightemperature insulating state, which has entropy of the orderof logð2Þ, populates many states of the plaquette withsignificant probability. In contrast, there is only one sig-nificant eigenvalue of the density matrix in low tempera-ture, corresponding to the singlet state. The insulatingphase at low temperature has thus very small entropy,and the bad metal has larger entropy; hence, decreasingtemperature favors insulator over metal. The actual firstorder line (dashed line in Fig. 1(a) inside the coexistenceregion, where the free energy of the two phases equals)therefore bends back and criticalU decreases with decreas-ing temperature. It is apparent that the zero temperaturetransition in cluster DMFT happens at Uc1 and not at Uc2

as in DMFT.

While the shape of the DMFT phase diagram stronglyresembles the phase diagram of the Cr-doped V2O3, thereentrant shape of the cluster-DMFT transition resemblesmore the !-organic diagram [17] as pointed out in Ref. [4].To understand the effects brought about by the short

range magnetic correlations near the transition, we focuson the local spectral functions displayed in Fig. 2. As insingle-site DMFT, below Uc1 [Fig. 2(a)] the system is anormal Fermi liquid with a reduced width of the quasipar-

-0.2 -0.1 0.0 0.1 0.2r t/U

0.00

0.05

0.10

0.15

0.20

0.25

t/T

Fermiliquid

Paramagneticinsulator

Bad

metal

Bad

insulator

b)

single site

FIG. 1 (color online). (a) The phase diagram of the paramag-netic half-filled Hubbard model within plaquette CDMFT. Inset:The histogram of the two insulating states. It shows the proba-bility for a given cluster eigenstate among the 16 eigenstates ofthe half filled plaquette. The singlet plaquette ground state hasthe highest probability. (b) For comparison, the correspondingphase diagram of the single-site DMFT (using the same 2Ddensity of states) is shown. The coexistence region is shown asthe shaded region. The dashed line marks the crossover abovethe critical point. The crossover line was determined by thecondition that the imaginary part of the self-energy at few lowestMatsubara frequencies is flat at the crossover value of U. Foreasier comparison, the x axis is rescaled and the reduced value ofUr ¼ U%UMIT

UMITis used. The critical value of U is UMIT ¼ 6:05t in

the cluster case and UMIT ¼ 9:35t in the single-site case.Pentagons in panel (a) mark the points in phase diagram forwhich we present the local spectral functions in Fig. 2.

PRL 101, 186403 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

31 OCTOBER 2008

186403-2




as in DMFT.



-0.2 -0.1 0.0 0.1 0.2r t/U

0.00

0.05

0.10

0.15

0.20

0.25

t/T

Fermiliquid


Bad

metal

Bad

insulator

b)

single site





31 OCTOBER 2008

186403-2

• DMFT phase diagram(paramagnetic)!

• Mott insulator at T=0: macroscopic entropy L log2!

• superexchange does not lift the ground-state degeneracy!

• missing feedback of nonlocal(magnetic) correlations!

• consequence: at T>0 the high entropy stabilizes the insulatorF = E-TS

Mott transition: cellular DMFT




as in DMFT.



-0.2 -0.1 0.0 0.1 0.2r t/U

0.00

0.05

0.10

0.15

0.20

0.25

t/T

Fermiliquid


Bad

metal

Bad

insulator

b)

single site





31 OCTOBER 2008

186403-2




as in DMFT.



-0.2 -0.1 0.0 0.1 0.2r t/U

0.00

0.05

0.10

0.15

0.20

0.25

t/T

Fermiliquid


Bad

metal

Bad

insulator

b)

single site





31 OCTOBER 2008

186403-2




as in DMFT.



-0.2 -0.1 0.0 0.1 0.2r t/U

0.00

0.05

0.10

0.15

0.20

0.25

t/T

Fermiliquid


Bad

metal

Bad

insulator

b)

single site





31 OCTOBER 2008

186403-2




as in DMFT.



-0.2 -0.1 0.0 0.1 0.2r t/U

0.00

0.05

0.10

0.15

0.20

0.25

t/T

Fermiliquid


Bad

metal

Bad

insulator

b)

single site





31 OCTOBER 2008

186403-2

Park et al. (2008)!

• DMFT correct at high temperatures: nonlocal correlations attenuated!• the transition line Uc(T) bends back at low T!• Uc lower by roughly a factor two!• nonlocal (short-range) correlations allow for nonlocal singlet formation!• unique ground state of the Mott insulator!

nonlocal correlations: DCALocal order and the gapped phase of the Hubbard model: A plaquette dynamical mean-field investigation

-8 -6 -4 -2 0 2 4 6 8

Energy [t]0

0.1

0.2

0.3

DO

S

Atot(ω)AP(ω)Mean Field AtotMean Field AP

Fig. 2: Solid line: on-site spectral function computed bymaximum entropy analytical continuation of QMC data forU = 6t and doping x= 0. Dashed line: spectral function in theP = (0,π), (π, 0)-momentum sector. Dotted and dash-dottedlines: P = (0,π), (π, 0) and local spectral functions obtainedby performing the DCA momentum averages of the standardSDW mean-field expressions for the Green function, with gap∆= 1.3t.

N = 1 and U ! 10t G(0,β/2) increases as T decreases,indicating the development of a coherent Fermi-liquidstate. In the 4-site DCA results a transition is evident as Uis increased through U∗ ≈ 4.2t: for U <U∗ A(0) increasesslowly as T is decreased, as in the single-site model, but forU >U∗, A(0) decreases, signalling the opening of a gap.The very rapid change across U =U∗ is consistent witha first-order transition, as found in the careful CDMFTanalysis of Park et al. [21]. The critical U is seen tobe essentially independent of temperature indicating thatthe entropies of the metallic and non-metallic states arevery similar. The end point of the first-order transitionis at about T = 0.25t which is approximately the Neeltemperature of the single-site method, at U = 4t [23].Figure 2 shows as the solid line the local electron spec-

tral function computed by maximum entropy analyticalcontinuation of our QMC data for U = 6t and n= 1.Analytical continuation is well known to be an ill-posedproblem, with very small differences in imaginary timedata leading in some cases to very large differences in theinferred real-axis quantities. A measure of the uncertain-ties in the present calculation comes from the differencebetween the spectra in the positive-energy and negative-energy regions, which should be equal by particle-holesymmetry. We further note that the gap is consistent withthe behavior shown in fig. 1. The local spectral functionexhibits a characteristic two-peak structure found alsoin CDMFT calculations [21]. The dotted line gives thespectral function for the Px-sector, corresponding to anaverage of the physical spectral function over the region(π/2< px < 3π/2), (−π/2< py < π/2); this is seen to bethe origin of the gap-edge structure.

0.1 0.2 0.3 0.4 0.5T [t]

0.1

0.12

0.14

Dou

ble

Occ

upan

cy

plaquette1 site AFM1 site PM

0.1 0.2 0.3 0.4 0.5T [t]

0.01

0.02

0.03

0.04

0.05

Dou

ble

Occ

upan

cyplaquette1 site AFM1 site PM

Fig. 3: Temperature dependence of double occupancy ⟨n↑n↓⟩computed using the 1-site and 4-site DCA methods as afunction of temperature for the half-filled Hubbard model atU = 5t (upper panel) and U = 10t (lower panel). The 1-sitecalculations are done for both paramagnetic and antiferro-magnetic phases, whereas the 4-site calculation is done for theparamagnetic phase only.

We present in fig. 3 the temperature dependence of thedouble-occupancy D= ⟨n↑n↓⟩ computed using the 1-siteand 4-site DCA for a relatively weak and a relativelystrong correlation strength. In the single-site approxi-mation antiferromagnetic correlations are absent in theparamagnetic phase and become manifest below the Neeltemperature; the difference between paramagnetic andantiferromagnetic phases therefore gives insight into thephysics associated with the antiferromagentic correlations.For the weaker interaction strength U = 5t, the develop-ment of Fermi-liquid coherence as T is decreased in theparamagnetic phase means that the wave function adjuststo optimize the kinetic energy, thereby pushing the inter-action term farther from its extremum and increasing D.At this U the magnetic transition is signaled by a rapiddecrease in D, indicating that the opening of the gapenables a reduction of interaction energy, as expected ifSlater physics dominates. For the larger U = 10t in the

37009-p3

• Mott gap or Slater gap ?!• k = (pi,0), (0.pi): Mott (pole of self-energy)

k = (0,0), (pi,pi): Slater-type transition!• k-selective transition

Gull et al. (2008)!• paramagnetic insulator!• plaquette, Lc=4

note: o.b.c. = p.b.c.still C-DMFT DCA!

• QMC solver, MaxEnt!• four-peak structure

cf. lattice-QMC results !!• low-energy peaks:

nonlocal correlations!• consistent with AF

Hartree-Fock calculations!• HF: LRO -> doubling of unit

cell, gap opening at the boundary of the reduced BZ

6=<latexit sha1_base64="meIe2rSWYcOySpHCZzG7KFban04=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48VTVtoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fikrZNMMfRZIhLVDalGwSX6hhuB3VQhjUOBnXByO/c7T6g0T+SjmaYYxHQkecQZNVZ66EscVGtu3V2ArBOvIDUo0BpUv/rDhGUxSsME1brnuakJcqoMZwJnlX6mMaVsQkfYs1TSGHWQL06dkQurDEmUKFvSkIX6eyKnsdbTOLSdMTVjverNxf+8XmaimyDnMs0MSrZcFGWCmITM/yZDrpAZMbWEMsXtrYSNqaLM2HQqNgRv9eV10m7UPbfu3V/Vmo0ijjKcwTlcggfX0IQ7aIEPDEbwDK/w5gjnxXl3PpatJaeYOYU/cD5/AEHTjbc=</latexit><latexit sha1_base64="meIe2rSWYcOySpHCZzG7KFban04=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48VTVtoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fikrZNMMfRZIhLVDalGwSX6hhuB3VQhjUOBnXByO/c7T6g0T+SjmaYYxHQkecQZNVZ66EscVGtu3V2ArBOvIDUo0BpUv/rDhGUxSsME1brnuakJcqoMZwJnlX6mMaVsQkfYs1TSGHWQL06dkQurDEmUKFvSkIX6eyKnsdbTOLSdMTVjverNxf+8XmaimyDnMs0MSrZcFGWCmITM/yZDrpAZMbWEMsXtrYSNqaLM2HQqNgRv9eV10m7UPbfu3V/Vmo0ijjKcwTlcggfX0IQ7aIEPDEbwDK/w5gjnxXl3PpatJaeYOYU/cD5/AEHTjbc=</latexit><latexit sha1_base64="meIe2rSWYcOySpHCZzG7KFban04=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48VTVtoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fikrZNMMfRZIhLVDalGwSX6hhuB3VQhjUOBnXByO/c7T6g0T+SjmaYYxHQkecQZNVZ66EscVGtu3V2ArBOvIDUo0BpUv/rDhGUxSsME1brnuakJcqoMZwJnlX6mMaVsQkfYs1TSGHWQL06dkQurDEmUKFvSkIX6eyKnsdbTOLSdMTVjverNxf+8XmaimyDnMs0MSrZcFGWCmITM/yZDrpAZMbWEMsXtrYSNqaLM2HQqNgRv9eV10m7UPbfu3V/Vmo0ijjKcwTlcggfX0IQ7aIEPDEbwDK/w5gjnxXl3PpatJaeYOYU/cD5/AEHTjbc=</latexit><latexit sha1_base64="meIe2rSWYcOySpHCZzG7KFban04=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48VTVtoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fikrZNMMfRZIhLVDalGwSX6hhuB3VQhjUOBnXByO/c7T6g0T+SjmaYYxHQkecQZNVZ66EscVGtu3V2ArBOvIDUo0BpUv/rDhGUxSsME1brnuakJcqoMZwJnlX6mMaVsQkfYs1TSGHWQL06dkQurDEmUKFvSkIX6eyKnsdbTOLSdMTVjverNxf+8XmaimyDnMs0MSrZcFGWCmITM/yZDrpAZMbWEMsXtrYSNqaLM2HQqNgRv9eV10m7UPbfu3V/Vmo0ijjKcwTlcggfX0IQ7aIEPDEbwDK/w5gjnxXl3PpatJaeYOYU/cD5/AEHTjbc=</latexit>

DCA

HF

diagrammatic extension of DMFT

Schäfer et al. (2015)!• dynamical vertex approx.

(and lattice QMC)!• gapped spectra for all U

(due to scattering from extended AF fluctuations)!

• no metal-insulator transition!• crossover from Slater to

Heisenberg physics(but no symmetry breaking,consistent with MW66)

• is this the final answer?!• diagrammatic approaches:

there is no control parameter (like Lc)

unconventional superconductivityVCA to the high-TC problem3

FIG. 1: AF (bottom) and dSC (top) order parameters forU = 8t as a function of the electron density (n) for 2 × 3,2 × 4 and 10-site clusters. Vertical lines indicate the firstdoping where only dSC order is non-vanishing.

FIG. 2: AF and dSC order parameters as a function of theelectron density on a 2 × 4 cluster for U = 6t, U = 8t andU = 12t.

large, n = 0.87 to n = 0.93 for L = 10 for example. Un-fortunately, finite cluster effects do not allow us to obtainreliable results for larger dopings for either the hole orelectron cases. The dSC order parameter is not presenton bonds between clusters, making the results more sen-sitive to size effects than in the case of the AF phase.Fig. 1 also shows, in the electron-doped case, a AF+dSCphase where AF and dSC order parameters are both non-vanishing [18, 19, 20]. We verified that, as expected fromsymmetry, the π-triplet order parameter is non-vanishingin that phase[21, 22], which is separated from a pure dSCphase by a quantum critical point around 13% doping,

near the value suggested by experiment [23, 24]. The sit-uation is less clear on the hole-doped side where the L = 6cluster has a very small doping range for the AF+dSCphase, the L = 8 cluster a large one, while the L = 10cluster shows none. This suggests that the way in whichthe AF and dSC phases approach each other on the hole-doped side cannot be accurately described by the smallvariational space that we use. Additional order parame-ters, such as stripe [2] or checkerboard orders [3] observedin certain cuprates may be necessary to get the full pic-ture. No SO(5) symmetric point [25] appears in our cal-culation in a size-independent way. On the other hand,our results for D0 in Fig. 1 show unambiguously thatthe pure dSC phase appears over a much broader rangeof dopings for hole- than for electron-doped cuprates, asobserved experimentally.

It is also instructive to know how the ground-state or-der parameters vary with interaction strength U , espe-cially because several normal-state calculations for thepseudogap [26, 27] show that the interaction strength forelectron-doped cuprates near optimal doping should be inthe weak to intermediate coupling range (U ∼ 6t), withU increasing as n decreases. A look at Fig. 2 for D0 andM0 shows that the range of dopings where only D0 isnon-vanishing is larger on the hole than on the electron-doped side for all values of U . That range increases withU in all cases so that a drop in U as n increases rein-forces the electron-hole difference in the size of the puredSC region. The range where only the dSC order pa-rameter is finite nearly doubles in going from U = 6t toU = 12t but the maximum value of D0 decreases, at leaston the hole-doped side. A stronger repulsion allows sys-tems with more holes to be superconducting, but at thesame time suppresses superconductivity more effectivelycloser to half-filling. Note that the dSC order parameterD0 should not be confused with the critical temperature:The maximum in D0 that appears near n = 0.9 on thehole-doped side does not mean that the maximum Tc isaround that doping. Instead, the maximum comes fromthe growth of D0 towards half-filling until proximity withthe AFM phase makes it fall rapidly [22]. Finally, as seenin previous calculations, as U increases from 6t to 12t, theAF phase does not extend as far on the electron-dopedside, [28] whereas the value of the order parameter M0

at half-filling increases [29].Fig. 3 shows intensity plots of the spectral functions at

the Fermi level, for U = 8t, in the first quadrant of theBrillouin zone. The left illustrates a hole-doped systemin a pure dSC phase. The spectral weight is concentratedalong the diagonal. This is observed even without long-range order [26], but is also compatible with the vanish-ing of the dSC gap along the diagonal. On the right, wedisplay an electron-doped system in a AF+dSC phase.The spectral weight is depleted along the diagonal andconcentrated near the zone boundaries ((π, 0) and (0, π)).This is also observed in the absence of long-range order

Senechal et al. (2004)!

AF order

parameter

dSC

order

parameter

electron density n

Senechal et al. (2005)!

• T=0, Lc=6,8,10, VCA(“simplified C-DMFT”)U=8, t=1, t’=-0.3, t’’=0.2!

• spontaneous symmetry breaking with nonlocal order parameter (not accessible to DMFT)!

• SC coexists with AF!• pure SC at higher hole

doping levels!• artificial finite-size and

cluster-geometry effects

• are cluster approaches efficient to solve the high-Tc problem ?

conclusions

DMFTuseful!beautiful!requires extensions

cluster !extensions

diagrammatic extensions

not systematic!respect symmetries!

can be motivated physically

systematic!limited in practice!artificial symmetry breaking and restoration

conclusions

cluster extensionssystematic!limited in practice!artificial symmetry breaking and restoration!not unique

C-DMFT DCAreal-space perspective k-space perspective

convergence with Lc = L1D: for local observables: exponential!for extended observables: 1/L1

for extended observables: 1/L12

respects translational symmetries!SSB: must be anticipated

breaks translational symmetries!SSB: “automatic”

conclusions

cluster extensionswhich type of physics is dominated !by local and short-range correlations?!

there is a DMFT scenario!cluster approaches: this is incorrect in D=2!

cluster approaches themselves not reliable in D=2 ?!

Mott transition

high-Tc superconductivitycannot be explained within DMFT!cluster approaches: short-range correlations are essential!likely: longer-range correlations important as well

conclusions

cluster extensions

systematic!limited in practice!

artificial symmetry breaking and restoration!

efficient cluster solver: QMC only (?)!systems with nonlocal or even long-range interactions there is no straightforward real-space cluster DMFT !

more issues

THE END.

Documents

Cluster Extensions of Dynamical Mean-Field Theory · For a small cluster and likewise for a system of disconnected clusters, even for the interacting case, it is comparatively simple