Quantum Geometry, Exclusion Statistics, and the Geometry of “Flux Attachment” in 2D Landau levels

“Reduced Density Matrices in Quantum Physics and Role of Fermionic Exchange Symmetry” :workshop Pauli2016 on Oxford University, April 12-15 2016

F. Duncan M. HaldanePrinceton University

 The degenerate partially-filled 2D Landau level is a remarkable environment in which kinetic energy is replaced by "quantum geometry” (or an uncertainty principle) that quantizes the space occupied by the electrons  quite differently from the atomic-scale quantization by a periodic arrangement of atoms. In this arena, when the short-range part of the Coulomb interaction dominates, it can lead to  “flux attachment”, where a particle (or cluster of particles) exclusively occupies a quantized region of space. This principle underlies both the incompressible fractional quantum Hall fluids and the composite-fermion Fermi liquid states that occur in such systems.

When it describes a “quantum geometry”

Q:

• In this case space is “fuzzy”(non-commuting components of the coordinates), and the Schrödinger description in real space (i.e., in “classical geometry”) fails, though the Heisenberg description in Hilbert space survives

• The closest description to the classical-geometry Schrödinger description is in a non-orthogonal overcomplete coherent-state basis of the quantum geometry.

A:

When is a “wavefunction” NOT a wavefunction?

Schrödinger vs Heisenberg

• Schrödinger’s picture describes the system by a wavefunction 𝝍(r) in real space

Erwin_schrodinger1.jpg (JPEG Image, 485 × 560 pixels) - Scaled (71%) http://www.camminandoscalzi.it/wordpress/wp-content/uploads/2010/09/Erwin_sc...

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Werner Karl Heisenberg

per la fisica 1932

Werner Karl HeisenbergDa Wikipedia, l'enciclopedia libera.

Werner Karl Heisenberg (Würzburg, 5dicembre 1901 – Monaco di Baviera, 1º febbraio1976) è stato un fisico tedesco. Ottenne il PremioNobel per la Fisica nel 1932 ed è considerato unodei fondatori della meccanica quantistica.

Indice

1 Meccanica quantistica2 Il lavoro durante la guerra3 Bibliografia

3.1 Autobiografie3.2 Opere in italiano3.3 Articoli di stampa

4 Curiosità5 Voci correlate6 Altri progetti7 Collegamenti esterni

Meccanica quantisticaQuando era studente, incontrò Gottinga nel 1922. Ciò permise lo sviluppo di unafruttuosa collaborazione tra i due.

Heisenberg ebbe l'idea della , la prima formalizzazione dellameccanica quantistica, nel principio di indeterminazione, introdotto nel 1927,afferma che la misura simultanea di due variabili coniugate, come posizione e quantità dimoto oppure energia e tempo, non può essere compiuta senza un'incertezza ineliminabile.

Assieme a Bohr, formulò l' della meccanica quantistica.

Ricevette il Premio Nobel per la fisica "per la creazione della meccanicaquantistica, la cui applicazione, tra le altre cose, ha portato alla scoperta delle formeallotrope dell'idrogeno".

Il lavoro durante la guerraLa fissione nucleare venne scoperta in Germania nel 1939. Heisenberg rimase inGermania durante la seconda guerra mondiale, lavorando sotto il regime nazista. Guidò ilprogramma nucleare tedesco, ma i limiti della sua collaborazione sono controversi.

Rivelò l'esistenza del programma a Bohr durante un colloquio a Copenaghen nelsettembre 1941. Dopo l'incontro, la lunga amicizia tra Bohr e Heisenberg terminòbruscamente. Bohr si unì in seguito al progetto Manhattan.

Si è speculato sul fatto che Heisenberg avesse degli scrupoli morali e cercò di rallentare ilprogetto. Heisenberg stesso tentò di sostenere questa tesi. Il libro Heisenberg's War diThomas Power e l'opera teatrale "Copenhagen" di Michael Frayn adottarono questainterpretazione.

Nel febbraio 2002, emerse una lettera scritta da Bohr ad Heisenberg nel 1957 (ma maispedita): vi si legge che Heisenberg, nella conversazione con Bohr del 1941, non espressealcun problema morale riguardo al progetto di costruzione della bomba; si deduce inoltreche Heisenberg aveva speso i precedenti due anni lavorandovi quasi esclusivamente,convinto che la bomba avrebbe deciso l'esito della guerra.

Sito web per questa immagineWerner Karl Heisenbergit.wikipedia.org

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• Heisenberg’s picture describes the system by a state ｜𝝍⟩ in Hilbert space

• They are only equivalent if the basis of states in real-space are orthogonal:

|ri

(r) = hr| i hr|r0i = 0(r 6= r0)

requiresthis failsin a quantumgeometry

• Schrödinger’s real-space form of quantum mechanics postulates a local basis of simultaneous eigenstates |x⟩ of a commuting set of projection operators P(x), where P(x)P(x′) = 0 for x ≠ x′.

Schrödinger vs Heisenberg and quantum geometry

(x) = hx| i| iHeisenberg Schrödinger

?=

only equivalent ifhx|x0i = 0 for x 6= x

0 this fails in aquantum geometry

• In “classical geometry” particles move from x to x’ because they have kinetic energy

• In “quantum geometry”, they move because the states |x⟩ and |x’⟩ are not only non-orthogonal, but overcomplete:

In this case the positive Hermitian operator

S(x,x0) = hx|x0i has null eigenstatesX

x

0

S(x,x0) (x0) = 0

(so the basis cannot be reorthogonalized)

• If the Schrödinger basis is on a lattice, so |x⟩ is normalizable

d(x,x0)2 = 1� |S(x,x0)|2 = 0= 1 x 6= x

0x = x

0

(trivial distance measure)

In this case kinetic energy (Hamiltonian hopping matrix elements) sews the lattice together

• In a quantum geometry there is a non-trivial Hilbert-Schmidt distance between (coherent) states on different lattice sites, and the Hamiltonian appears “local”

Hilbert-Schmidt distance

H =X

x

V (x)|xihx| hx|x0i 6= �(x,x0)

• Fractional quantum Hall effect in 2D electron gas in high magnetic field (filled Landau levels)

1/3L =

Y

i<j

(zi � zj)3Y

i

e�|zi|2/4`2B

• Laughlin (1983) found the wavefunction that correctly describes the 1/3 FQHE , and got Nobel prize,

⌫ = 13

• Its known that it works, (tested by finite-size numerical diagonalization) but WHY it works has never really been satisfactorily explained!

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3 3.5

laughlin 10/30

E

Collective mode with short-range V1 pseudopotential, 1/3 filling (Laughlin state is exact ground state in that case)

“roton”

(2 quasiparticle + 2 quasiholes)

goes intocontinuum

gap incompressibility0 1 20

0.5

klB

Moore-Read ⌫ = 24

“kF ”

fermionic“roton”

bosonic “roton”

Collective mode with short-range three-body pseudopotential, 1/2 filling (Moore-Read state is exact ground state in that case)

• momentum ħk of a quasiparticle-quasihole pair is proportional to its electric dipole moment pe ~ka = �abBpbe

k�B

gap for electric dipole excitations is a MUCH stronger condition than charge gap: doesn’t transmit pressure!

fractional-charge, fractional statistics vortices

Chiral edge states at edge of finite droplet of fluid

(Halperin, Wen)

time

e2i�ei�

charge -e/m

statisticsθ= π/m

e.g., m=3

=Y

i,↵

(zi � w↵)Y

i<j

(zi � zj)mY

i

e� 1

4`2B

z⇤i zi

• Non-commutative geometry of Landau-orbit guiding centers

⇥e�

O

~r

~R

~Rc

~R

~r

~Rc

displacement of electron from origin

displacement of guiding center from origin

displacement of electron relative to guiding center of Landau orbit

shape of orbit around guiding center is fixed by the cyclotron effective mass tensor

~r = ~R+ ~Rc

[Rx, Ry] = �i`2B

[Rx

c

, Ry

c

] = +i`2B

[rx, ry] = 0classical geometry

Landau orbit(harmonic oscillator)

[Ra, R

bc] = 0 (a, b 2 {x, y})guiding centers commute with Landau radii

quantum geometry

guidingcenter

The one-particle Hilbert-space factorizes

• FQHE physics is *COMPLETELY* defined in the many-particle generalization (coproduct) of

H = ¯HGC ⌦ Hc

[Rx

c

, Ry

c

] = +i`2B

[Rx, Ry] = �i`2B

space isomorphicto phase space in whichthe Landau orbit radii act

space isomorphicto phase space in whichthe guiding-centers act

HGC

Once is discarded, the Schrödinger picture is no longer valid! Hc

~r = ~R+ ~Rcclassical electron coordinate

• Laughlin states also occur in the second Landau level, and in graphene, and more recently in simulations of “flat-band” Chern insulators

Previous hints that the Laughlin “wavefunction” should not be interpreted as a wavefunction:

These don’t fit into the original paradigm of the Galileian-invariant

Landau level

• First, translate Laughlin to the Heisenberg picture:

a = 12z + @z⇤

a† = 12z

⇤ � @z

Landau-levelladder operators

0(z, z⇤) = e�

12 z

⇤z

a 0(z, z⇤) = 0

Guiding-center ladder operators

a = 12 z + @z⇤

a† = 12 z

⇤ � @zz $ z

z = z⇤

a 0(z, z⇤) = 0

a†f(z) 0(z, z⇤) = zf(z) 0(z, z

⇤)

a† = 12z � @z⇤ a = 1

2z⇤ + @z

Gaussian lowest-weight state

action of guiding-center raising operators on LLL states

usual identification is

• Heisenberg form of Laughlin state (not “wavefunction”)

| 1/qL i =

0

@Y

i<j

(a†i � a†j)q| 0i

1

A⌦ (| 0i)

ai| 0i = 0 ai| 0i = 0

2 HGC ⌘ H

2 Hc ⌘ H

Guiding-centerfactor (keep)

Landau-orbit factor (discard)

• At this point we discard the Landau-orbit Hilbert space.

• The only “memory” of the shape of the Landau orbits is “hidden” in the definition of a

• guiding-center Coherent states (single particle)a(g)| g(0)i = 0

| g(z)i = eza†(g)�z⇤a(g)| g(0)i

• This is a non-orthogonal overcomplete basisS(z, z0) = h g(z)| g(z

0)i• non-zero eigenvalues of the positive Hermitian

overlap function are holomorphic!Z

dz0dz0⇤

2⇡S(z, z0) (z0, z0⇤) = (z, z⇤)

(z, z⇤) = f(z⇤)e�12 z

⇤z

The “purified” Laughlin state

• This is now defined in the many-particle guiding-center Hilbert space, without reference to any Landau-level structure

• What defines ?

| 1/qL i =

Y

i<j

(a†i � a†j)q| 0i ai| 0i = 0

L(g) =gab2`2B

X

i

RaiR

bi

[L(g), a†i (g)] = a†i (g)It is the raising

operator for the “guiding-center spin”

L(g)of particle i

a†i

gab is a 2x2 positive-definite unimodular (det = 1) 2D spatial metric tensor

• The Laughlin state has suddenly revealed its well-kept secret- a hidden geometric degree of freedom! It is parameterized by a unimodular metric gab!

| 1/qL (g)i =

Y

i<j

(a†i (g)� a†j(g))q| 0(g)i

ai(g)| 0(g)i = 0

• In the naive LLL wavefunction picture, the unimodular metric gab is fixed to be proportional to the cyclotron effective mass tensor m*ab.

• In the reinterpretation it is a free parameter.

This is the entire problem:nothing other than this matters!

• generator of translations and electric dipole moment!

H =X

i<j

U(Ri �Rj)

[Rx, Ry] = �i`2B

[(Rx

1 �Rx

2), (Ry

1 �Ry

2)] = �2i`2B

• relative coordinate of a pair of particles behaves like a single particle

• H has translation and inversion symmetry

[(Rx

1 +Rx

2), (Ry

1 �Ry

2)] = 0

[H,P

iRi] = 0

two-particle energy levels

like phase-space, has Heisenberg uncertainty principle

gap

want to avoidthis state

• Laughlin state

U(r12) =⇣A+B

⇣(r12)

2

`2B

⌘⌘e� (r12)2

2`2B 0

E2 symmetric

antisymmetric

• Solvable model! (“short-range pseudopotential”) 12 (A+B)

12B

rest all 0

| mL i =

Y

i<j

⇣a†i � a†j

⌘m|0i

ai|0i = 0 a†i

=Rx + iRy

p2`

B

EL = 0

maximum density null state

• m=2: (bosons): all pairs avoid the symmetric state E2 = ½(A+B)

• m=3: (fermions): all pairs avoid the antisymmetric state E2 = ½B[ai, a

†j ] = �ij

• New feature is similar to FQH ferromagnet, where electrons couple to a combination of magnetic flux and Berry curvature of the ferromagnetic order parameter(Skyrmions)

• The electron density is no longer rigidly tied to the magnetic flux density, it can deviate from it at the expense of paying the correlation energy cost for geometric distortion.

• Old results of Girvin, Macdonald and Platzman (O(q4) “guiding-center structure factor”) get a simple explanation as zero-point fluctuations of the geometry

• The key idea for understanding both the Fractional Quantum Hall and Composite Fermi Liquid states is “Flux attachment”

• quantum solid

• repulsion of other particles make an attractive

potential well strong enough to bind particle

• unit cell is correlation hole

• defines geometry

solid melts if well is not strong enough to contain zero-point motion (Helium liquids)

• similar story in FQHE:

• “flux attachment” creates correlation hole

• potential well must be strong enough to bind electron

• defines an emergent geometry

• new physics: Hall viscosity, geometry............

e-

• continuum model, but similar physics to Hubbard model

but no broken symmetry

(�1)p ⇥ (�1)pq = +1exchange of p fermions

Berry phase(exchange of

“exclusion zones”)

compositeis a boson

Statistical selection rule

the rule formerly known as “odd-denominator”, (but Moore-Read has p=2, q=4)

• elementary unit of the FQHE fluid with ν= p/q is a “composite boson” of p electrons that exclude other electrons from a region with q London (h/e) flux quanta

p=1, q=3⅓ Laughlin

⅓ Laughlin(with different shape)

p=2, q=5 ⅖ Hierarchy/Jain

central orbitaloccupied

next twoempty

central two orbitals occupied, next three empty

ν= ⅓

ν= ⅓

ν= ⅖

compositesexchange as

bosons

“exclusion statistics”

• The metric (shape of the composite boson) has a preferred shape that minimizes the correlation energy, but fluctuates around that shape

• The zero-point fluctuations of the metric are seen as the O(q4) behavior of the “guiding-center structure factor” (Girvin et al, (GMP), 1985)

• The metric has a companion “guiding center spin” that is topologically quantized in incompressible states.

�E / (distortion)

2

L= 12

32

52

⅓⅓ ⅓1 0 0

32

12

configuration of “elementary droplet” (composite boson)

subtract total L (=Lref) of reference configuration(uniform occupation p/q)

totalL ⅓ Laughlin

s = ( 12 � 32 ) = �1

• Origin of FQHE incompressibility is analogous to origin of Mott-Hubbard gap in lattice systems.

• There is an energy gap for putting an extra particle in a quantized region that is already occupied

• On the lattice the “quantized region” is an atomic orbital with a fixed shape

• In the FQHE only the area of the “quantized region” is fixed. The shape must adjust to minimize the correlation energy.

e-

energy gap prevents additional electrons from entering the

region covered by the composite boson

• The usual “lowest Landau level wavefunction” formalism has

(x) = f(z)e�14 z

⇤z/`2B

holomorphicfunction

• With a (quasi) periodic boundary condition, this becomes

(z, z⇤) /

N�Y

i=1

�(z � wi)

!e� 1

4z⇤z`2B

X

i

wi = 0

Weierstrasssigma function* zeroesN�

(one for each flux quantum passing through the primitive region of the pbc)*(slightly modified from Weierstrass’ original definition

when the pbc lattice is not square or hexagonal)

• In the Heisenberg-algebra reinterpretation X

i

wi = 0

• The filled Landau level is

| i =

0

@Y

i<j

�(a†i � a†j)�(P

ia†i )

1

A |0i

• The Laughlin states are

| i =

0

@Y

i<j

�(a†i � a†j)m

1

AmY

k=1

�(P

ia†i � wk)|0i

mX

k=1

wk = 0.

| i =N�Y

i=1

�(a†i � wi)|0i one particle

filled LevelN = N�

N = 1

N� = mNLaughlin state⌫ = 1

m

• Unlike the filled Landau level state, in which the only metric-dependence is the normalization, the Laughlin states depend on the metric choice which fixes the shape of the vortex-like correlation hole around each particle (“attached flux”)

| i =

0

@Y

i<j

�(a†i � a†j)m

1

AmY

k=1

�(P

ia†i � wk)|0i

⌫ = 1m

correlation holesin two states with different metrics

(filled Landau level is a Slater-determinant state with no correlation hole)

x

“flux attachment”Has a shape that defines a metric

δR

displacement of charge relative to center offlux attachment gives

an electric dipole

momentum

flux attachment creates a correlation hole that can bind one or more particles into a composite object

"(P , g)correlation energy dispersion

p particles+ q “flux”(orbitals)

“kinetic energy” = electric polarization energy

(velocity

a) =

@"

@pa

pa = B✏ab(e�Rb)

• The key idea is that (at the correct particle density) the Berry phase from motion of the attached vortex cancels the Bohm-Aharonov phase from motion of the charge

• This means the Lorentz force is canceled by the Magnus force, and the composite object moves in straight lines like a neutral particle

Bosonscan condense in the p = 0 (inversion-symmetric) state with no electric dipole

Fermionscan form a Fermi sea in “momentum” (dipole)space

• exchange phase

p particles + q “flux” (orbitals)

(�1)pq⇠p

-1 for electrons

= +1 composite objectis boson

= −1 composite objectis fermion

e.g., one electron with p = 1, q = 2

• inversion symmetry of FQHE : gcd(p,q) = 1 or 2

Berry curvature of the “Flux attachment” of a vortex-like correlation hole modifies the statistics

e

the electron excludes other particles from a region containing 3 flux quanta, creating a potential well in which it is bound

1/3 Laughlin state If the central orbital is filled, the next two are empty

The composite bosonhas inversion symmetry

about its center

It has a “spin”

.....

.....−1 0 013

13

13

12

32

52

L = 12

L = 32−

s = �1

(composite boson picture)

(that couples to Gaussian curvature of its metric)

e

2/5 hierarchy/Jain state

e.....

.....−

1 0

12

32

52

−

0 0125

25

25

25

25

L = 2

L = 5

s = �3

L =gab2`2B

X

i

RaiR

bi

Qab =

Zd2r rarb�⇢(r) = s`2Bg

ab

second moment of neutral composite boson

charge distribution

(composite boson picture)

Jain’stwo filled $ $ “ -levels?”�

• choose distinct “occupied orbitals” (allowed dipole moments, quantized by the pbc){di, i = 1, . . . N} 2 { L

N }

which minimize

for fixed d =1

N

X

i

di

1

N

X

i<j

|di � dj |2 =1

2

X

i

|di � d|2

Model for 1/m CFL states

• is a many-body quantum number that takes N2 distinct values. There is thus one such configuration per sector of this many-body translational quantum number.

¯d mod { LN

}

• The model 1/m CFL states ( including the boson case m = 1) are

({zi, z⇤i }, {di}, {w↵}) /✓deti,j

Mij({zk}, dj , d⇤j , d)◆

⇥

0

@Y

i<j

�(zi � zj)

1

Am�2

mY

↵=1

�((P

izi)� w↵)NY

i=1

e� 1

4

z⇤i zi

`2B

Mij({zk}, dj , d⇤j , d) = e1m

d⇤j zi

2`2B

Y

k 6=i

�(zi � zk � dj + d)

• The matrix in the determinant is

• also:mX

↵=1

w↵ =NX

j=1

dj = Nd

F. D.M.H and E. H. Rezayi, unpublished; (m=2 case given in Shao et al, PRL 114, 206402 (2015)

complex cf dipoles edj

(dj is quantized in units )L

N

mean value of dj

• Now we see that the “Fermi sea” is invariant under uniform translation in “dipole space”

py

px

filledempty

cluster of adjacent occupied states

pbc 2⇡~L

0.028677091503 0 1110101000 0.0286770915235 0 0001010111

0.0171543754946 0 1110110000 0.0172785391733 0 0001001111

0.00272205658268 0 11110000010.00272205658096 0 00001111100.00741749061239 0 11110000100.00741749061624 0 00001111010.0131254758865 0 1111000100 0.0131430302064 0

00001110110.0172785391469 0 1111001000 0.017154375511 0

00001101110.0141743825022 0 1111010000 0.0141743825338 0

00001011110.00547651410185 0 11111000000.00547651412427 0 0000011111

# Z_{COM} overlap with PH-conjugate in opposite charge sector 1-overlap0 0.999998870263 1.1297367517e-061 0.999999369175 6.3082507884e-072 0.99999860296 1.39704033186e-063 0.99999860296 1.3970403312e-064 0.999999369175 6.30825078063e-075 0.999998870263 1.12973675237e-066 0.999999369175 6.30825079173e-077 0.99999860296 1.39704032942e-068 0.99999860296 1.39704032909e-069 0.999999369175 6.30825078507e-07

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model state is numerically very close to p-h symmetry

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something related to “generalized Pauli constraints”

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O(q4)

Imprint of shape-fluctuations of the metric

(GMP)

Pij =1

N�

X

q

0eiq·(Ri�Rj)

S(1) = ⌫(1± ⌫)

FDMH arXiv:1112.0990

fermions

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S(1) = ⌫(1± ⌫)

FDMH arXiv:1112.0990

⇠ = ±1

bosons

fermions

because of Pauli, function is its own Fourier transform!

2

The interaction also has rotational invariance if

v(q) = v(qg), q2g ≡ gabqaqb, (6)

where gab is the inverse of a positive-definite unimodular(determinant = 1) metric gab; this will only occur if theshape of the Landau orbits are congruent with the shapeof the Coulomb equipotentials around a point charge onthe surface. In practice, this only happens when there isan atomic-scale three-fold or four-fold rotation axis nor-mal to the surface, and no “tilting” of the magnetic fieldrelative to this axis, in which case gab = ηab.I will assume that translational symmetry is unbroken,

so ⟨c†αcα′⟩ = νδαα′ , where ν is the “filling factor” of theLandau level. (In a 2D system, this will always be trueat finite temperatures, but may break down as T → 0).Then ⟨ρ(q)⟩ = 2πνδ2(qℓB). Note that the fluctuationδρ(q) = ρ(q) − ⟨ρ(q)⟩ also obeys the algebra (4). I willdefine a guiding-center structure factor s(q) = s(−q) by

12 ⟨{δρ(q), δρ(q

′)}⟩ = 2πs(q)δ2(qℓB + q′ℓB). (7)

This is a structure factor defined per flux quantum, and isgiven in terms of the GMP structure factor s(q) of Ref.[1](defined per particle) by s(q) = νs(q). I also define sa(q)≡ ∂s(q)/∂qa, sab(q) ≡ ∂2s(q)/∂qa∂qb, etc.In the “high-temperature limit” where |v(r)| ≪ kBT

for all r, but kBT remains much smaller than the gapbetween Landau levels, the guiding centers become com-pletely uncorrelated, with ⟨c†αcα′c†βcβ′⟩ − ⟨c†αcα′⟩⟨c†βcβ′⟩

→ s∞δαβ′δβα′ , with s∞ = ν + ξν2, where ξ = −1if the particles are spin-polarized fermions, and ξ =+1 if they are bosons (which may be relevant forcold-atom systems). Note that for all temperatures,limλ→∞ s(λq) = s∞, while s(0) = limλ→0 s(λq) =kBT/

!

∂2f(T, ν)/∂ν2"

"

T

#

, where f(T, ν) is the free en-ergy per flux quantum. s(0) vanishes at T = 0, and atall T if v(λq) diverges as λ → 0; the high temperatureexpansion at fixed ν, for rq ≡ eaϵabqbℓ2B, is

s(q)− s∞(s∞)2

= −

$

v(q) + ξv(rq)

kBT

%

+O

$

1

T 2

%

. (8)

The correlation energy per flux quantum is given by

ε =

&

d2qℓ2B4π

v(q) (s(q)− s∞) . (9)

The fundamental duality of the structure function (al-ready apparent in (8), and derived below) is

s(q)− s∞ = ξ

&

d2q′ℓ2B2π

eiq×q′ℓ2B (s(q′)− s∞) . (10)

This is valid for a structure function calculated us-ing any translationally-invariant density-matrix, and as-sumes that no additional degrees of freedom (e.g., spin,valley, or layer indices) distinguish the particles.

Consider the equilibrium state of a system with tem-perature T and filling factor ν with the Hamiltonian(2). The free energy of this state is formally given byF [ρeq], where ρeq(T, ν) is the equilibrium density-matrixZ−1 exp(−H/kBT ) and F [ρ] is the functional

F [ρ] = Tr (ρ(H + kBT log ρ)) , (11)

which, for fixed ν, is minimized when ρ = ρeq. TheAPD corresponding to a shear is Ra → Ra + ϵabγbcRc,parametrized by a symmetric tensor γab = γba. Let ρ(γ)= U(γ)ρeqU(γ)−1, where U(γ) is the unitary operatorthat implements the APD, and F (γ) ≡ F [ρ(γ)] = F [ρeq]+ O(γ2), which is is minimized when γab = 0. The freeenergy per flux quantum has the expansion

f(γ) = f(T, ν) + 12G

abcd(T, ν)γabγcd +O(γ3), (12)

where Gabcd = Gbacd = Gcdab. The “guiding-center shearmodulus” (per flux quantum) of the state is given by Gac

bd= ϵbeϵdfGaecf , with Gac

bd = Gcadb, and Gac

bc = 0. (Notethat in a spatially-covariant formalism, both stress σa

b(the momentum current) and strain ∂cud (the gradientof the displacement field) are mixed-index tensors thatare linearly related by the elastic modulus tensor Gac

bd.)The entropy is left invariant by the APD, and the onlyaffected term in the free energy is the correlation energy,which can be evaluated in terms of the deformed struc-ture factor s(q, γ), given by

s∞ + ξ

&

d2q′ℓ2B2π

eiq×q′ℓ2B (s(q′)− s∞)eiγabqaq

′

bℓ2

B , (13)

with γab ≡ ϵacϵbdγcd. This gives Gabcd(T, ν)γabγcd as

γabγcdϵaeϵcf

&

d2qℓ2B4π

v(q)qeqfsbd(q;T, ν). (14)

Assuming only that the ground state |Ψ0⟩ of (2) hastranslational invariance, plus inversion symmetry (so ithas vanishing electric dipole moment parallel to the 2Dsurface), GMP[1] used the SMA variational state |Ψ(q)⟩∝ ρ(q)|Ψ0⟩ to obtain an upper bound E(q) ≤ f(q)/s(q)to the energy of an excitation with momentum !q (orelectric dipole moment eeaϵabqbℓ2B), where

f(q) =

&

d2q′ℓ2B4π

v(q′)!

2 sin 12q × q′ℓ2B

#2s(q′, q),

s(q′, q) ≡ 12 (s(q

′ + q) + s(q′ − q)− 2s(q′)) . (15)

Other than noting it was quartic in the small-q limit,GMP did not not offer any further interpretation of f(q).It can now be seen to have the long-wavelength behavior

limλ→0

f(λq) → 12λ

4Gabcdqaqbqcqdℓ4B, (16)

and is controlled by the guiding-center shear-modulus.Then the SMA result is, at long wavelengths, at T = 0,

E(q)s(q) ≤ 12G

abcdqaqbqcqdℓ2B. (17)