Topological spin liquids with PEPSnqs2014.ws/archive/Presen...Topological spin liquids with PEPS Didier Poilblanc Laboratoire de Physique Théorique, Toulouse • Some motivations:

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

jeudi 13 novembre 14

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


http://arxiv.org/abs/1308.3463

http://arxiv.org/abs/1308.3463

http://arxiv.org/find/cond-mat/1/au:+Poilblanc_D/0/1/0/all/0/1


http://arxiv.org/find/cond-mat/1/au:+Schuch_N/0/1/0/all/0/1


http://arxiv.org/find/cond-mat/1/au:+Cirac_J/0/1/0/all/0/1












http://arxiv.org/find/cond-mat/1/au:+Perez_Garcia_D/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Perez_Garcia_D/0/1/0/all/0/1







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)


The many-body spectrum of a topological liquid

{

EExcitations are fractional (created by pairs)

Degeneracy from «topological order»


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:


Materials realizing a spin-1/2 Kagome QAFM

e.g. experiments by C. Broholm (MIT), Z. Hiroi (ISSP), P. Mendels (Orsay)...


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 ?


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


http://ojps.aip.org/getabs/insref_abs.jsp?key=PRBMDO&prog=getinsref&id=710361&idtype=inspec

http://ojps.aip.org/getabs/insref_abs.jsp?key=PRBMDO&prog=getinsref&id=710361&idtype=inspec

Two «types» of cylinder boundaries

«even» «odd»Gv = +1 Gv = �1

s s̄

cylinder geometry


Eigenstates of a «Wilson loop» operator

no vison flux:

Z2 vison flux:

+�

wv = +1

wv = �1

cylinder geometry


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


HAKLT =X

hi,ji

PSi+Sj=4ij

S=2... vs «Trivial» insulator

The GS is unique !

The Affleck-Kennedy-Lieb-Tasaki spin liquid


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


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


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


⌦ |

↵ «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


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


Simple rewriting for Kagome lattice :

map on a square lattice

«defect triangle»dphys = 8


Improving the RVB / PEPS ...

«defect triangles» cost energy !

dressing by introducing NNN singlets

(Zeng & Elser 90’)

The «Simplex RVB»


Easy formalism to construct all topological GS !

(+)e

(�)e

(+)o

(�)o

orthogonal in the limit of infinite cylinders(Nh = 1)

= Z


diag blocks

Off-diagonal blocks

«mixed» TM

Misguich et al. PRL ‘02

! 1

< 1

RK RVB


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


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)


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...


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...


http://arxiv.org/find/cond-mat/1/au:+Wang_L/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Wang_L/0/1/0/all/0/1



http://arxiv.org/find/cond-mat/1/au:+Gu_Z/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Gu_Z/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Wen_X/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Wen_X/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Verstraete_F/0/1/0/all/0/1


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:

=


RVB spin liquid

doping

X

VB

X

VB+holes

Mott insulator Superconductor ?

+

+

· · ·

2nd step :

i

c⇥

i


hole kinetic energy⌦c†i,�cj,�

↵


�ij =⌦ci,"cj,#

↵

c = 0

c = �0.65

Superconducting pairing amplituded+is symmetry

(almost) pure d-wave symmetry


Variational energy : fPEPS vs iPEPS

iPEPS

fPEPS

Very good ansatz !


BA

«Holographic» framework

Reduced density matrix

�A = TrB |�⇥�

�|

bulk

edge?


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)










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}-


Numerical results

STE ' � ln 2 Z2 spin liquid


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�


LT

Short-range entanglement Hamiltonian

Trivial gapped spin liquid

A(r) ⇥ exp (�r/�b)

AKLT cylinders(D=2)

⇠b traks bulk correlation length

· · ·


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


gapless ES !

Entanglement spectrum

should reflect the physical edge

spectrumLi & Haldane conjecture


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 !


Documents

Topological spin liquids with PEPSnqs2014.ws/archive/Presen...Topological spin liquids with PEPS Didier Poilblanc Laboratoire de Physique Théorique, Toulouse • Some motivations: