Bulk-Boundary Correspondence
Tensor Networks

Conference on „Quantum Matter, Space-Time, and Quantum Information“
Yukawa Institute for Theoretical Physics,
University of Kyoto, June 14th, 2016

TENSOR NETWORKS

MANY-BODY QUANTUM SYSTEMTENSOR NETWORK STATES
i1

Ψ, ρSpins (fermions)

Efficient description guided by entanglement
Represent wide range of physical behavior
Algorithms

i2i3

OUTLINE
Bulk-boundary correspondence in PEPS
JIC, Poilblanc, Schuch, and Verstraete, PRB 83, 245134 (2011)
JIC, D. Perez-Garcia, N. Schuch, F. Verstraete, arxiv:1606.00608

| Ψ〉

AρA

Edge theories in PEPS
Yang, Lehman, Poilblanc, Acoley, Verstraete, JIC, Schuch, PRL 112, 036402 (2014)… …

…

PROJECTED ENTANGLED-PAIR STATES (PEPS)

PEPS

Entanglement swappinga)

|Ψ1>

=| φ〉

|Ψ1>

|Φ>

D

∑ | n, n 〉n =1b)

|Ψ1>

|Ψ1>

|Φ>

PEPS
Verstraete, JIC (2004)

|Ψ1>

|ΨΝv>

…
|Ψ2>

=|Ψνv,2>… …

… …
=|ΨNv,Nh>

… …

…

PEPS
Verstraete, JIC (2004)

PLANE

… …

CYLINDER

…

… …
TORUS

… …

…

…

PEPS

|Ψ1>

… … …

…

| Ψ〉

Other geometries and topologies

Provide efficient descriptions for local theories
Molnar, Schuch, Verstraete, JIC (2015)

PEPS
Verstraete, JIC (2004)

|Ψ1>

… … …

…

| Ψ〉

PEPS as a tensor network:=| Ψ1 〉

∑Ai
Aαβγδ

i
αβγδ

| i; α, β, χ, δ〉Tensor network

Easy to handle

PEPS
AREA LAW
Area law:tr [| Ψ〉〈Ψ |]ρ=AA

S ( ρ A )  N ∂A

All PEPS fulfilll area law: S ( ρ A )  N ∂A log DA

Aboundary

D N∂A

PEPS
PARENT HAMILTONIANS
Torus:PARENT HAMILTONIAN:=H| Ψ〉∑h ≥ 0nhLocal

… … …

…

Frustration-free:hn | Ψ〉 = 0h

Ground state:
H | Ψ〉 = 0

GappeddV
D N∂ASYMMETRIESTOPOLOGYSYMMETRIESGLOBAL
Wolf, Perez-Garcia, Sanz, Verstraete, JIC (2008)
Perez-Garcia, Wolf, Gonzalez, JIC (2010)
Singh, Vidal (2012)

… … …

…

vg

ug

vg†

= vg
vg†

u g⊗ N | Ψ〉 = | Ψ〉g ∈Gvg

ug

=ugvg

vg†

vg

vg†
vg†

SYMMETRIESLOCAL GAUGE
Tagliacozzo, Celi, Lewenstein (2014)
Haegeman, van Acoley, Schuch, JIC, Verstraete (2015)
Zohar, Walh, Burrello (2015)

… … …

…

|Ψ1>

=u g | Ψ1 〉 vg ⊗ 1 ⊗ vg† ⊗ 1 | Ψ1 〉 =vg ⊗ 1 ⊗ 1 ⊗ vg† | Ψ1 〉 =u g⊗4 | Ψ〉 = | Ψ〉g ∈Gg ∈G| Ψ4 〉=u g⊗4 | Ψ 4 〉 = vg vg† ⊗ ... ⊗ ... | Ψ 4 〉 =| Ψ 4 〉TOPOLOGY
Schuch, JIC, Perez-Garcia (2010)
Zauner et al (2014)

… … …

…
vgvg†

vgvg†

g ∈G=

TOPOLOGYSTRINGS
Schuch, JIC, Perez-Garcia (2010)
Zauner et al (2014)=+| Ψ Aa 〉| Ψ Bb 〉| Ψ 〉| Ψ 〉 = 〈φab | S | Ψ Aa 〉 | Ψ Bb 〉S = vg⊗C

TOPOLOGYSTRINGS
Schuch, JIC, Perez-Garcia (2010)
Zauner et al (2014)

vgvg†

vg
vg†

Closed strings: Ground state

Ground state degeneracy
Locally indistinguishability

Open strings: Excitations
Braiding
Quantum computationBULK-BOUNDARY CORRESPONDENCE
JIC, Poilblanc, Schuch, and Verstraete, PRB 83, 245134 (2011)
JIC, D. Perez-Garcia, N. Schuch, F. Verstraete, arxiv:1606.00608

BULK-BOUNDARY CORRESPONDENCE
Poilblanc, Schuch, Verstreate, JIC (2011)

|Ψ1>

A| Ψ〉

A
ρA

σ ∂A = U † ρ AU

BULK-BOUNDARY CORRESPONDENCE
Poilblanc, Schuch, Verstreate, JIC (2011)

|Ψ1>

AA
ρA| Ψ〉σ ∂A = U † ρ AUPROOFA

A

Polar decomposition: M = UQ=UQBULK-BOUNDARY CORRESPONDENCE
SYMMETRIES & TOPOLOGY| Ψ〉AρAσ ∂A = U † ρ AU

Symmetries:u g⊗ N | Ψ〉 = | Ψ〉σ ∂A = vg⊗ N †σ ∂Avg⊗ N∂A∂Ag ∈GTopology:⊗ N ∂A †σ ∂A v=σ ∂A σ ∂Avg⊗ N∂A=gRestricts the subspace: topological correction to the area law

BULK-BOUNDARY CORRESPONDENCE
FIXED POINTS RENORMALIZATIONCirac, Perez-Garcia, Schuch, Vestraete (2016)

local commuting−Hσ ∂A e==
P Pe − HCorse-grainσ ∂ANon-local projector(topology)

RENORMALIZATIONFIXED POINTS
Verstraete, Latorre, Rico, Wolf, JIC (2004)

Equivalence relation: | Ψ〉 ≈| Φ〉
Two states are equivalent if they are connected by a local transformation
| Ψ〉

| Φ〉

Renormalization: Mapping among equivalence classes

MATRIX PRODUCT STATESFIXED POINTS RENORMALIZATION
Verstraete, Latorre, Rico, Wolf, JIC (2004)
Vidal (2005)

Fixed points:
Classes: connected by a (local) unitary
Can be locally disentangled

U†U

MATRIX PRODUCT STATESFIXED POINTS RENORMALIZATIONCirac, Perez-Garcia, Schuch, Vestraete (2016)

PURE STATESMIXED STATES

UTUS

†

Zero correlation length
Saturate the area law for mutual information

EDGE THEORIES
Yang, Lehman, Poilblanc, Acoley, Verstraete, JIC, Schuch, PRL 112, 036402 (2014)

EDGE THEORY

Torus:PARENT HAMILTONIAN:
H=|Ψ>∑h ≥ 0nhLocal
Frustration-free:|Ψ1>hn | Ψ〉 = 0

Ground state:
H | Ψ〉 = 0hSpectrum:
E1E0∆

EDGE THEORY

Spin system with a physical boundary
PARENT HAMILTONIAN:
H=∑h ≥ 0nhLocal
|Ψ1>

… … …

…

Frustration-free:hn | Ψ〉 = 0h

Ground state:
H | Ψ〉 = 0

Spectrum:
E1

Degenerate E0∆

EDGE THEORY

Spin system with a physical boundary

h|Ψ1>

… … …

…
=0h

Isometry

hPolardecomposition=0boundary D N∂A=Positive

EDGE THEORYLOW ENERGY EXCITATIONSIsometry:=0

Maps bulk-boundary (cf AdS/CFT)Pastawski, Yoshida, Harlow, Preskill (2015)
Hayden, Nezami, Qi, Thomas, Walther, Yang (2016)

Characterizes the GS subspace of H,

H | Ψ〉 = 0EDGE THEORYLOW ENERGY EXCITATIONS
Local perturbation in the bulk: H ' =H + ε V =H + ε∑vnhDegenerate perturbation theory==H edge ε P0 ∑vn P0n∑ hα β | Ψα 〉〈Ψ β | α ,β ,α 〉〈 β | H=
∑ hα β |= α ,β ,boundary

VIt can be interpreted as a Hamiltonian acting on the boundary
The isometry maps the bulk operator onto the boundary

EDGE THEORYLOW ENERGY EXCITATIONS
Local perturbation in the bulk: H ' =H + ε V =H + ε∑vnhSpectrum
E1E0edge states

Global and gauge symmetries are preserved
Topology gives rise to a local projector: superselection rule at the boundary

EDGE THEORYMODELAKLT model + bulk perturbations

EDGE THEORYCONCLUSIONS & OUTLOOKPEPS offer efficient descriptions of many-body quantum systems
Enconde symmetries and topology in a simple way

BULK-BOUNDARYEDGESAρAσ ∂A = U † ρ AU|Ψ1>
Isometry maps bulk operators onto the boundary
Symmetries (global & gauge) are inherited
Topology appear as a superselection rule (anomaly)