Entanglement Spectrum and Boundary Theories

in topological spin liquids

Didier Poilblanc

Laboratoire de Physique Théorique, Toulouse

COLLABORATORS

* Ignacio Cirac (Max-Planck Garching)

* Norbert Schuch(Cal. Tech. -> RWTH Aachen)

* Frank Vertraete (Univ. Vienna)

* David Perez-Garcia(Univ. Madrid)

- Phys. Rev. B 83, 245134 (2011) [1,2,3]- Phys. Rev. B 86, 014404 (2012) [1,2,4]- Phys. Rev. B 86, 115108 [1,2,4]- arXiv:1210.5601 [1,2,4]

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Topological order (and edge modes) in condensed matter correlated systems - from FQHS to quantum magnetism ?

Some warm-up: entanglement spectra of Heisenberg ladder

“Holographic mapping” from PEPS wavefunctions: connection between bulk and boundary

Application to 2d topological states: SU(2)-invariant RVB wavefunctions

OUTLINE

Exotic states of matter

Beyond the “order parameter paradigm”: correlations “missed” by two-point correlation functions can be detected by entanglement measures

* no broken symmetry* no local order * GS degeneracy depends on topology of space

X. G. WenTopological order

Exemple: (topological) spin liquid

Edge states in (topological) FQH systems

A

BLi & HaldanePRL 2008

Lauchli et al., 2009

L

L

Some correspondence between edge spectrum and entanglement spectrum

Regnault, Bernevig & Haldane, 2009

Entanglement spectrum : {�i}

rewrite the weights as: �i = exp (�⇥i/2)

Rewrite as thermal density matrix�A

⇥A =⇤

i

�2i |i

⇥A

�i|A

Boundary (pseudo-)hamiltonian ?⇠ = � ln (⇢A)

How does the boundary Hamiltonian reflect bulk properties ?

The simplest example:the 2-leg antiferromagnetic

spin “ladder”

A sub-sytem

Analytic formof the “boundary hamiltonien” :

�A = exp (�Hb)

D.P., PRL 105, 077202 (2010)Cirac et al., PRB 83, 245134 (2011)

Exponential decay !

Jleg = 0 � T� =⇥

Jrung = 0 � T� = 0

D.P., PRL 105, 077202 (2010)

Jrung = sin �

Jleg = cos �

Heisenberg ladderHb

Effective temperature

“bulk”edge

A B

- Extend to long cylinders with Nh legs ?Nh �⇥ ?

- Get a simple physical description of the degrees of freedom of Hb

BL BR

RL

PEPS

Tensor Network approaches

Matrix Product States (1D) :

|��

=⇥

I

cI |i1, i2, ..., iNh

�

....1 2 3 Nh

i

M i�1,�2 D�D matrix

Equivalent to DMRG !!D ~ m parameter controling the DMRG truncation

ik = �S,�S + 1, ... , S � 1, S

I. CiracF. Verstraete

G. Vidal

Romer and Ostlund (PRL, 1995)

cI =�

�

Li1�1

M i2�1�2

... MiNh�1�

Nh�2�Nh�1

RiNh�

Nh�1

= tr{Li1M i2 ... M iNh�1RiNh}

Totsuka and M. Suzuki., J. Phys.: Condens.Matter,7(1639), 1995.

Tensor Network for d=2 (and higher): Projected Entangled Paired States (PEPS)

“contract” product of tensors

Basic formula:

isometry: maps 2D onto 1D

“live” on the boundary:natural identification of

EMERGING degrees of freedom

Holographic framework

Obvious consequence: area law !

lA = Nh/2

L R

A B

�A L�L

R

R�⇢b

⇢A = ⇢B = U⇢b U†

Hb = � ln ⇢b

* gapped systems (AKLT): is short-range approaching a critical point (deformed AKLT): becomes long-range

* for topological GS (Kitaev toric code, spin liquids, etc...): does have special properties ? is it non-local ?

Hb

Hb

Hb

To what extend is a local Hamiltonian ?Hb

Topological spin liquids

{GS have different “winding

numbers”but same IRREP of space

group

E

Excitations are topological “visons”

Degeneracy from “topological order”

degeneracy depends on topology:g=4 on the torus

g=2 on the cylinder

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Boundary theories of topological Resonating Valence Bond states

PERMUTATION OF: Norbert Schuch,1, 2 David Perez-Garcıa,3 J. Ignacio Cirac,4 and Didier Poilblanc5

1Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena CA 91125, U.S.A.2Institut fur Quanteninformation, RWTH Aachen, D-52056 Aachen, Germany3Department of Mathematical Analysis, Faculty of Mathematics, UCM, Spain

4Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany5Laboratoire de Physique Theorique, C.N.R.S. and Universite de Toulouse, 31062 Toulouse, France

To be WRITTEN...

PACS numbers: 71.10.-w,75.10.Kt,03.67.-a,03.65.Ud

I. INTRODUCTION

In order to detect topological order, a common setupconsists of dividing the system into two regions (named Aand B) and compute the reduced density matrix (RDM)in the GS of e.g. the A subsystem. In particular, the en-tanglement entropy (EE), defined as the Von Neumannentropy of the RDM SVN = ��A ln �A, contains an exten-sive term – proportional to the length of the boundary(area law) – and a universal sub-leading constant, thetopological EE, characterizing the topological nature. Inaddition, � ln �A can be seen as a (dimensionless) Hamil-tonian Hb which provides (even) more information on thesystem. First, its spectrum, the so-called entanglementspectrum (ES), has been conjectured to have a deep cor-respondence with the actual boundary spectrum. Thisremarkable property was first established in fractionalquantum Hall states1 – the ES was shown to reproducefaithfully the spectrum of edge states – and, later on, inquantum spin systems2. Furthermore, beyond its spec-trum, the nature of Hb itself is directly linked to theproperty of the bulk. PEPS o�er a natural formulationof the relation between bulk and boundary. In Ref. 3, anexplicit isometry was constructed which maps the Hamil-tonian Hb onto another one Hb acting on the space ofauxiliary spins living at the boundary of region A, whilekeeping the spectrum. Furthermore, for various two-dimensional (2D) models displaying quantum phase tran-sitions, like a deformed AKLT4 or an Ising-type5 model,it was found3 that a gapped bulk phase with local ordercorresponds to a boundary Hamiltonian with local inter-actions, whereas critical behavior in the bulk is reflectedon a diverging interaction length of Hb.

II. RVB WAVEFUNCTIONS ON CYLINDERS

Here, to investigate the boundary Hamiltonian Hb ofthe RVB wavefunction, we consider cylinders of lengthNh and circumference Nv, as depicted in Fig. ??(a). Par-titioning the cylinder into two half-cylinders (playing therole of the two A and B subsystems defined above) revealstwo edges L and R along the cut, as shown in Fig. ??(a,b).Ultimately, we will take the limit of infinite cylinders, i.e.Nh ⇥ ⇧. As we shall see later, for a topological state,

the boundary Hamiltonian Hb depends on the choice ofthe boundaries BL and BR. Open boundary conditions(OBC) on the cylinder ends (Fig. 1(a)) are obtained bysetting the outgoing virtual indices to “2” as shown inFig. 1(b). Arbitrary boundary conditions can be realizedas in Fig. 1(c,d).

FIG. 1: (Color online) Cylinder geometry (Nh = 4) used tocompute the RDM. Equal-weight superposition of hardcore-dimer coverings [see e.g. (a,c)] have simple representations interms of PEPS (b,d). The bipartition generates two L andR edges along the cut. Various (fixed) boundary conditionsBL and BR can be chosen on the cylinder ends by fixing theboundary (virtual) variables. OBC (a) are defined by settingall boundary indices to “2” (b). Arbitrary boundary condi-tions can be defined physically by freezing (with e.g. a localmagnetic field) some spins at the boundaries (c), translatingin the PEPS language by setting the boundary indices to 0(spin ⇥) or 1 (spin �) (d).

PEPS representation of RVB states. – We start withthe square lattice RVB wavefunction (NN | ⇤⌅⌃ � | ⌅⇤⌃

Resonating Valence Bond states

RVB = equal-weight superposition of NN singlet coverings D=3 PEPS

F. Verstraete et al., 2006P. Fazekas and P.W. Anderson Philosophical Magazine 30, 423-440 (1974)

RVB on the kagome lattice

map on a square lattice (but no reflection symmetry)

Evidence for Z2 liquid from recent numerics:

PEPS:

Yan, Huse & White, Science 2011

* Square lattice: algebraic dimer-dimer correlations Albuquerque & Alet, PRB 2010

* Kagome lattice: short-range dimer-dimer correlations

Some properties of (short-range) RVB wavefunctions

Misguich et al., PRL 2002Moessner & Sondhi, PRL 2001

Yang & Yao, PRL 2012

gapped Z2 spin liquids

i

j

j i j

(a) (b)i ji

Disconnected topological sectorsin the space of dimer lattice coverings

On a kagome cylinder: even and odd sectors

4� 2 cylinders

P = Peven

(kagome)

P = Podd

Topological sectors translate into CONSERVATION LAWS of “transfer matrix”

Structure of the Boundary Hamiltonian

supported by the non-zero eigenvalue sector of the RDM

non-universal partTC: H

local

= 0

Topological universal part

Hb = � ln ⇢b

Numerics of Boundary Hamiltonian(infinite cylinders)

Hlocal is an extended t-J model !

local operator (on the edge):D �D matrix ⇥ basis of D2 operators

1

2

⌦ 0 representation

Topological entropy

SVN = S0 + ANv

# = 3Nv/2SVN ⇥ � ln #

# of states on the edge contributing to even sector:

(even)

gapless ES !

Entanglement spectrum

should reflect the physical edge spectrum

but might not generic of liquids...Z2

Interpolation between Kitaev’s TC and RVB states

can be mapped onto Kitaev’s toric code

gapless ES !

Conclusion and outlook

* natural mapping between bulk and boundary one-to-one correspondence between property of bulk and property of the boundary Hamiltonian * applied also to TOPOLOGICAL spin states tools to identify spin liquids in microscopic models E

??

ADDITIONAL MATERIALOn the topological energy splitting

Numerical data (Nh = 1)

YC20

Topological energy scaling: quantitative fits

... of practical use in DMRG calculations ? 400-sites YC12: �E�+ ⇠ 0.35 600-sites YC16: �E�+ ⇠ 0.07 900-sites YC20: �E�+ ⇠ 0.014 below S=0 and S=1 gaps !