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Topological spin liquids with PEPS
Didier Poilblanc
Laboratoire de Physique Théorique, Toulouse
• Some motivations: «trivial» vs «topological» spin liquids (on frustrated lattices)
• Simple physical ansatze using Projected Entangled Pair States
• «Holographic» framework : bulk-edge correspondence, entanglement Hamiltonian & detecting topological order from the PEPS
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COLLABORATORS
Norbert Schuch, DP, J. Ignacio Cirac, and David Pérez-García, Phys. Rev. B 86, 115108 (2012)DP, Norbert Schuch, David Pérez-García, and J. Ignacio Cirac, Phys. Rev. B 86, 014404 (2012)
DP, Norbert Schuch, J. Ignacio Cirac, Phys. Rev. B 88, 144414 (2013)DP and Norbert Schuch, Phys. Rev. B 87, 140407 (2013)
Norbert Schuch, DP, J. Ignacio Cirac, and David Perez-GarciaPhys. Rev. Lett. 111, 090501 (2013)
DP, Philippe Corboz, Norbert Schuch, J. Ignacio Cirac, PRB 2014
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Exotic «spin liquids» beyond the
«order parameter» paradigm
Topological order can also be detected by entanglement measures !
* no spontaneous broken symmetry* no local order * Topological order
X. G. Wen
- Do they exist in materials ? in simple models ?
- How to detect them ?
GS degeneracy (depends on topology of space)
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The many-body spectrum of a topological liquid
{
EExcitations are fractional (created by pairs)
Degeneracy from «topological order»
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Best candidate : spin-1/2 Heisenberg QAF on the Kagome lattice !
S. Yan, D.A. Huse & S. White, Science 2011
S. Depenbrock, I.P. McCulloch & U. Schollwock, PRL 2012
topological features ?
~S1 · ~S2
1
2
S = 1/2
Numerical «evidence» (DMRG) for a (gapped) spin liquid:
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Materials realizing a spin-1/2 Kagome QAFM
e.g. experiments by C. Broholm (MIT), Z. Hiroi (ISSP), P. Mendels (Orsay)...
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Low-energy spectrum N=36 sites
Sindzingre & Lhuillier, EPL 2009
Exponential # of singlets
Lecheminant et al., PRB 97
Jiang et al., PRL 2008Finite spin gap ?
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RVB spin liquid:
S = 0
spin-1/2 RVB
Equal-weight superposition of NN singlet coverings
P. Fazekas and P.W. Anderson Philosophical Magazine 30, 423-440 (1974)
Topological liquidHasting-Oshikawa-LSM theorem
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Two «types» of cylinder boundaries
«even» «odd»Gv = +1 Gv = �1
s s̄
cylinder geometry
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Eigenstates of a «Wilson loop» operator
no vison flux:
Z2 vison flux:
+�
wv = +1
wv = �1
cylinder geometry
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s s̄
v v
Gv = +1
Gv = �1
Wv = +1
Wv = �1 same class asKitaev’s Toric Code (fixed point )
spin liquid : topological GS inserting «spinons» and «visons»
Z2
⇠ = 0
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HAKLT =X
hi,ji
PSi+Sj=4ij
S=2... vs «Trivial» insulator
The GS is unique !
The Affleck-Kennedy-Lieb-Tasaki spin liquid
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Projected Entangled Pair States (PEPS) construction
Ex.: the S=2 AKLT
|S↵= |01
↵� |10
↵
D=2 spin-1/2 virtual states
Project onto physical subspace :
local tensor
d = 2S + 1
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The spin-1/2 RVB can also be written as a PEPS !
(D=3)
Project onto physical subspace S=1/2 (d=2)
1/2� 0virtual states:
PEPS tensor
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The PEPS as a variational ansatz
I. CiracF. Verstraete
G. Vidal↵,� = {1, . . . , D}
dimension of auxilary (or virtual) space
i = {1, · · · , dphys}
“contract” product of tensors
Coefficients of the wavefunction
C{i1,1,...,iNv,Nh}
| ↵=
XCi1,i2,...,iN |i1, i2, ..., iN
↵
N = NvNh
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⌦ |
↵ «Transfer Operator»
Build «double layer» tensor network by contracting physical variables
if D small enough exact contractions possible...
Iterate product of TM’s to build infinite cylinder
D2Nv ⇥D2Nv matrix
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use simple PEPS ansatz to investigate physical (e.g. topological, etc...) properties
optimize PEPS to construct competitive ansatz for microscopic models
Two possible «routes» :
G. Vidal, T. Xiang, R. Orus, P. Corboz,... & many more
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Simple rewriting for Kagome lattice :
map on a square lattice
«defect triangle»dphys = 8
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Improving the RVB / PEPS ...
«defect triangles» cost energy !
dressing by introducing NNN singlets
(Zeng & Elser 90’)
The «Simplex RVB»
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Easy formalism to construct all topological GS !
(+)e
(�)e
(+)o
(�)o
orthogonal in the limit of infinite cylinders(Nh = 1)
= Z
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diag blocks
Off-diagonal blocks
«mixed» TM
Misguich et al. PRL ‘02
! 1
< 1
RK RVB
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Finite size scaling of RVB energy
YC16 YC12 YC8 cylinders
s s̄
v v
Gv = +1
Gv = �1
Wv = +1
Wv = �1
dynamical fluctuations of vison/spinon pairs
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Gs energy splittings (semi-log scale)
Very short coherence lengths
(Nh ! 1)
⇠ < 1 unit cell
YC20 infinite cylinder
(NN RVB)
(NN RVB)
(Simplex RVB)
(Simplex RVB)
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Holon doping: could a d-wave superconductor emerge (on the square lattice)
?
A Little digression:
What happens under doping the RVB state
?
Spinon doping might be relevant to describe field induced phases...
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RVB superconductors relevant for high-Tc superconductivity ?
Idea #2: magnetic frustration is the key parameter to stabilize a (RVB) spin liquid on the 2D square lattice Ling Wang, DP, Zheng-Cheng Gu, Xiao-Gang Wen, Frank Verstraete, PRL 111 037202 (2013)
Idea #1: the RVB spin liquid is the «parent» insulator of the high-Tc superconductor P.W. Anderson, T.M. Rice, etc...
jeudi 13 novembre 14
RVB spin liquid
0
X
VB
fermion repn
ii
ii
s-wave d+is-wavebosonic repn fermionic repn
for « fPEPS » see e.g. :Corboz, Evenbly, Verstraete, Vidal, PRA 2010
RVB spin liquid
X
VB
1st step:
=
jeudi 13 novembre 14
RVB spin liquid
doping
X
VB
X
VB+holes
Mott insulator Superconductor ?
+
+
· · ·
2nd step :
i
c⇥
i
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hole kinetic energy⌦c†i,�cj,�
↵
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�ij =⌦ci,"cj,#
↵
c = 0
c = �0.65
Superconducting pairing amplituded+is symmetry
(almost) pure d-wave symmetry
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Variational energy : fPEPS vs iPEPS
iPEPS
fPEPS
Very good ansatz !
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BA
«Holographic» framework
Reduced density matrix
�A = TrB |�⇥�
�|
bulk
edge?
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Basic formula:
�2b
�A = U⇥2bU†
isometry: maps 2D onto 1D
“lives” on the boundary
lA = Nh/2
L R
A B
�A L�L
J. Ignacio Cirac, DP, Norbert Schuch, Frank VerstraetePhys. Rev. B 83, 245134 (2011)
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Entanglement entropy
“area” law
Levin & Wen, 2006
Kitaev & Preskill, 2006
subleading correction to area law:topological entropy
SVN ⇠ CNv � lnD
(Von Neumann)
SVN = �Tr{⇢A ln ⇢A} = Tr{�b ln�b}-
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Numerical results
STE ' � ln 2 Z2 spin liquid
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Entanglement Hamiltonian (acting on the edge)
�2b = exp (�Hb)
Is it local ?
The spectrum of Hb is in one-to-one with the true edge spectrum !
Li & Haldane
Hb
local operator (on the edge):D �D matrix ⇥ basis of D2 operators
edge site operators :
x�
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LT
Short-range entanglement Hamiltonian
Trivial gapped spin liquid
A(r) ⇥ exp (�r/�b)
AKLT cylinders(D=2)
⇠b traks bulk correlation length
· · ·
jeudi 13 novembre 14
Entanglement Hamiltonian highly non-local
Topological gapped spin liquid
supported by the non-zero eigenvalue sector of the RDM
projector characterizing topological sectors
Exemple: Kagome RVB
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gapless ES !
Entanglement spectrum
should reflect the physical edge
spectrumLi & Haldane conjecture
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Summary & Outlook
• Simple and physical understanding of important correlated phases of matter like :
• topological and trivial SLs,
• unconventional SC, ...
• Chiral SLs ?
• Virtual degrees of freedom play «physical role» at boundary : edge states
THANK YOU !
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