TensorNetworks

ShinjiTakeda

LATTICE2016July26,2016,UniversityofSouthampton,UK

What’stensornetwork?

Tijkl

tensor:laJcepointindices:link

Ti

j

kl

What’stensornetwork?

AtargetquanNty(wavefuncNon/parNNonfuncNon)isrepresentedbytensornetwork

T Ti

j

kl

m

n

o

�

··· ,i,···· · · TijklTmnio · · ·

contracNon

⇔

Whytensornetworks?

Freefromthesignproblem

∵Probabilitydoesnotenter

Tensornetworkapproaches

Hamiltonian/Hilbertspace Lagrangian/Pathintegral

Quantummany-bodysystem

Classicalmany-bodysystem/pathintegralrep.ofquantumsystem

WavefuncNonofgroundstate/excitedstates

ParNNonfuncNon

VariaNonalmethod ApproximaNon,Coarsegraining

RealNme,Out-of-equilibrium,QuantumsimulaNon

UsefulinequilibriumsystemsufferingfromthesignprobleminMC(μ≠0,θ≠0,etc.)

DMRG,MPS,PEPS,MERA,… TRG,SRG,HOTRG,TNR,Loop-TNR,…

n  AnsatzofWavefuncNon(inTensornetworkrepresentaNon)n  VariaNonalmethodtoobtaingroundstatein1Dquantumsys.(gapped)n  Quantumentanglementistakenintoaccountn  ByusinginformaNoncompression(SVD),onecandrasNcallyreducethe#

ofparametersO(2N)→O(N)whilekeepingaccuracyreasonably

Hamiltonianapproach

|�� =�

s1,s2,...,sN

�s1,s2,....,sN |s1� � |s2� � · · · � |sN �

s1 s2 sN

· · ·

· · · · · · · · ·

2Nelements

MatrixProductState(MPS)originatedfromDMRGWhite1992,Schollwoeck2004

s|±� = ±|±�

si = ±

n  AnsatzofWavefuncNon(inTensornetworkrepresentaNon)n  VariaNonalmethodtoobtaingroundstatein1Dquantumsys.(gapped)n  Quantumentanglementistakenintoaccountn  ByusinginformaNoncompression(SVD),onecandrasNcallyreducethe#

ofparametersO(2N)→O(N)whilekeepingaccuracyreasonably

Hamiltonianapproach

|�� =�

s1,s2,...,sN

�s1,s2,....,sN |s1� � |s2� � · · · � |sN �

As : d� d matrix

2Nelements

2d2Nelements

tr [As11 As2

2 · · · AsNN ]

s1 s2 sN

Tensornetworkrep.ofwavefuncNon

A1 AN

MatrixProductState(MPS)originatedfromDMRGWhite1992,Schollwoeck2004

si = ±

informaNoncompression

:Matrixproduct

Hamiltonianapproachn  Schwingermodel,periodicBC,spectrum,DMRG

Byrnes,Sriganesh,Bursill&Hamer,PRD66(2002)013002

n  Schwingermodel,openBC,spectrum,MPSBanuls,Cichy,Cirac,Jansen,JHEP11(2013)158

n  Schwingermodel,T≠0,MPS+MPOBanuls,Cichy,Cirac,Jansen&Saito,PRD92(2015)034519,PRD93(2016)094512

n  2DSU(2),realNmedynamicsofstringbreaking,MPSKuehn,Zohar,Cirac&Banuls,JHEP07(2015)130

n  SchwingermodelwithNf≠1andμ≠0Kuehn,7/25(Mon.)17:25TD

n  FermionswithlongrangeinteracNonusingMPSSzyniszewski,today’sPoster

relatedwithhighenergyphysics

Testbench:Schwingermodel(QED2)

A Hamiltonian approach: TNS as ansatz for states

AtT=0,groundstateandlowlyingexcitaNonscanberepresentedasMPS

Vector mass gap

m/g MPS with OBC DMRG0 0.56421(9) 0.56419(4)0.125 0.53953(5) 0.53950(7)0.25 0.51922(5) 0.51918(5)0.5 0.48749(3) 0.48747(2)

Scalar mass gap

MPS with OBC SCE1.1279(12) 1.11(3)1.2155(28) 1.22(2)1.2239(22) 1.24(3)1.1998(17) 1.20(3)

• variaNonalsearch• reliablecontrolofsystemaNcerrorsperformed• asainreliableconNnuumlimit

Banuls,Cichy,Cirac,Jansen,Saito,JHEP11(2013)158;PoSLAT13,332

Sign problem is overcome:mulNflavorcasewithchemicalpotenNalfromStefanKühn’stalk

Lohmayer,NarayananPRD88(2013)105030

analyNcalm/g=0

MPS results

fromK.Cichy

AtfiniteT,thermalequilibriumstatesdescribedbyMPO

Banuls,Cichy,Cirac,Jansen,Saito,PoSLAT14,302;Phys.Rev.D92,034519;PoSLAT15,283;Phys.Rev.D93,094512

imaginaryNme(thermal)evoluNoncanbesimulatedwithMPO-MPStechniquesallerrorsourceshavebeencontrolled

observables

e.g.thermalevoluNonofchiralcondensateform/g=0

fullphysicalspace:notruncaNonofelectricflux

truncaNngtheelectricfluxperlinkupto(converged)Lcut

densityoperator

A Hamiltonian approach: TNS as ansatz for states fromK.Cichy

Lagrangianapproach

TensorRenormalizaNonGroup(TRG)

∋

Levin&NavePRL99,120601(2007)

①  RewriteZintensornetworkrepresentaNon

Z ��

{s}

e��H[s] =�

i,j,k,l,...

...T ijklTmnio...

ProceduresofTRG

NewdegreesoffreedomSpins

・AnalyNc・exact

s

s

s

s

s

s

s

s

s

willbeexplainedsoon

T T

TTT

T

T T T

il o

m

nj

k

e.g.2DIsingmodel

ProceduresofTRG

TT (1)

BlockingofTensor(likespin-blocking)

•  extracNngimportantinformaNonnumerically•  selecNonofinformaNonintroducesapproximaNon

①  RewriteZintensornetworkrepresentaNon②  CoarsegrainingTensor

ProceduresofTRG

①  RewriteZintensornetworkrepresentaNon②  CoarsegrainingTensor③  Repeatthecoarsegrainingandthenreducethe

numberoftensors,finallycomputeZbycontracNon

T (2)

T (1)T

ProceduresofTRG

①  RewriteZintensornetworkrepresentaNon②  CoarsegrainingTensor③  Repeatthecoarsegrainingandthenreducethe

numberoftensors,finallycomputeZbycontracNonZ �

�

i,j

T (n)ijij

T (2)

i

jTT (1)

forPeriodicBC

①Tensornetworkrep.Z �

�

{s}

e��H[s] =�

i,j,k,l,...

...TijklTmnio...

sx = ±1

x y

ixy

e�sxsy = cosh(�sxsy) + sinh(�sxsy)= cosh� + sxsy sinh�

= cosh�(1 + sxsy tanh �)

= cosh�1�

ixy=0

(sxsy tanh �)ixy

newd.o.f.

1)  ExpandBoltzmannweightasinHigh-Texpansion2)  IdenNfyinteger,whichappearsintheexpansion,as

newd.o.f.→indexoftensor

e.g.2DIsingmodel

①Tensornetworkrep.Z �

�

{s}

e��H[s] =�

i,j,k,l,...

...TijklTmnio...

�

���

T0000 T0001 T0010 T0011

T0100 T0101 T0110 T0111

T1000 T1001 T1010 T1011

T1100 T1101 T1110 T1111

�

��� =

�

���

1 0 0 tanh �0 tanh� tanh � 00 tanh� tanh � 0

tanh � 0 0 (tanh�)2

�

���

�2(cosh�)2

1)  ExpandBoltzmannweightasinHigh-Texpansion2)  IdenNfyinteger,whichappearsintheexpansion,as

newd.o.f.→indexoftensor 3)  Integrateoutspinvariable(oldd.o.f.)4)  Gettensornetworkrep.!

e.g.2DIsingmodel

T T

TTT

T

T T T

il o

m

nj

k

①Tensornetworkrep.Z �

�

{s}

e��H[s] =�

i,j,k,l,...

...TijklTmnio...

1)  ExpandBoltzmannweightasinHigh-Texpansion2)  IdenNfyinteger,whichappearsintheexpansion,as

newd.o.f.→indexoftensor 3)  Integrateoutspinvariable(oldd.o.f.)4)  Gettensornetworkrep.!

Basicprocedureiscommontofermion/otherbosonsystem

Bosonfieldincompactgroup➞characterexpansion➞character:newd.o.f.

Meuriceetal.,PRD88,056005(2013)

①Tensornetworkrep.

n  Sofar,wehavejustrewrisenZn  NextstepistocarryoutthesummaNonn  But,naïveapproachcosts∝22Vn  IntroduceapproximaNonandreducethecostwhile

keepinganefficiencybysummingimportantpartinZ

Z =�

..,i,j,k,l,m,n,o,..

· · ·TijklTmnio · · ·

①Tensornetworkrep.

Z =�

..,i,j,k,l,m,n,o,..

· · ·TijklTmnio · · ·

Coarsegraining(renormalizaNon,blocking)

n  Sofar,wehavejustrewrisenZn  NextstepistocarryoutthesummaNonn  But,naïveapproachcosts∝22Vn  IntroduceapproximaNonandreducethecostwhile

keepinganefficiencybysummingimportantpartinZ

②CoarsegrainingDecomposiNonoftensor Newd.o.f.

S1

S3

m T ijkl =�

m

(S1)jkm(S3)limT

SingularvaluedecomposiNon(SVD)

②CoarsegrainingDecomposiNonoftensor

S1

S3

m

Newd.o.f.

�1 � �2 � ... � 0

u, v : unitarymatrix

:singularvalues

T ijkl =�

m

(S1)jkm(S3)limT

Mab =�

m

uam�m(v†)mb

SingularvaluedecomposiNon(SVD)

②CoarsegrainingDecomposiNonoftensor

S1

S3

m

Newd.o.f.

�1 � �2 � ... � 0

approx.

u, v : unitarymatrix

Tensor(matrix)isapproximatedbylow-ranktensor=informaNoncompression

:singularvalues

T ijkl =�

m

(S1)jkm(S3)limT

Tijkl = M(kj),(il) =all�

m

u(kj),m�

�m ·�

�mv†m,(il) �Dcut�

m=1

(S1)jkm(S3)lim

Mab =�

m

uam�m(v†)mb

②Coarsegraining

S4

S2

S1

S3

m

m

T ijkl �Dcut�

m=1

(S1)jkm(S3)lim

T ijkl �Dcut�

m=1

(S2)klm(S4)ijm

T T

T T

S1

S2S3

S4

S4

S2

S1

S3

②CoarsegrainingMakingnewtensorbycontracNon

S1

S2S3

S4

a

bc

d

T new

a

bc

d

integrateoutoldd.o.f.

all�

i,j,k,l

(S1)ila(S2)jib(S3)kjc(S4)lkd = T newabcd

=

RenormalizaNon-like!

TrT

2DIsingmodelonsquarelaJce

Dcut=32

0

0.5

1

1.5

2

2.5

3

3.5

1.6 1.8 2 2.2 2.4 2.6 2.8 3

Spec

ific

heat

T

L=4L=16L=64

L=1024

onlyonedayuseofthisMBA

Cost∝log(LaJcevolume)×(Dcut)6×[#temperaturemesh]

Tc = 2/[ln(1 +�

2)]= 2.269...

numericalderivaNve

C =1L2

�

�T

�T 2 � lnZ

�T

�

StatusofLagrangianapproachn  2Dsystem

n  Spinmodel:IsingmodelLevin&Nave2007,X-YmodelMeuriceetal.PRE89,013308(2014),X-YmodelwithFisherzeroMeuriceetal.PRD89,016008(2014),O(3)modelUnmuth-Yockeyetal.LATTICE2014,X-Ymodel+μMeuriceetal.PRE93,012138(2016)

n  Abelian-HiggsBazavovetal.LATTICE2015n  φ4theoryShimizuMod.Phys.Les.A27,1250035(2012)n  QED2,QED2+θShimizu&KuramashiPRD90,014508(2014)&PRD90,074503(2014)n  Gross-Neveumodel+μST&YoshimuraPTEP2015,043B01n  CP(N-1)+θKawauchi&STPRD93,114503(2016),Kawauchi7/25(Mon)17:05TDn  TowardsQuantumsimulaNonofO(2)modelZouetal,PRA90,063603n  TRGandquantumsimulaNonMeurice7/26(Tue.)15:40TD

n  3Dsystem,HigherorderTRG(HOTRG):acoarsegrainingmethodapplicableforanydimensionalsystemXieetal.PRB86,045139(2012)n  3DIsing,PossmodelWanetal.CPL31,070503(2014)n  3DFermionsystemSakai7/26(Tue.)18:10TD

n  DecoratedtensornetworkrenormalizaNonWisrichetal.NewJ.Phys.18,053009(2016)

ApplicaNontoCP(N-1)+θ

�

�0

ConNnuumlimit

� � 1/g2

1stPT

ToymodelofQCDStrongCPproblemSchierholtz1994,Ple�aetal.1997,Imachietal.1999,Ple�aetal.1999

2�

�

Seiberg1984Strongcouplinglimit

ApplicaNontoCP(1)+θ

�

�0

1stPT

2�

�

?2ndPT

Haldane’sconjecture:massgapofO(3)vanishesatθ=π≅CP(1)

O(3)+θwasintensivelystudiedbyMCandHaldane’sconjectureisconfirmedUniversalityclassisalsoconsistentwitheachotherBietenholtzetal.1995,Wieseetal.2012,deForcrandetal.2012,AzcoiNetal.2012,Allesetal.2014

Haldane1983

� � 1/g2

ApplicaNontoCP(1)+θ

�

�0 � � 1/g2

1stPT

2�

�

CP(1)MC,imaginaryθAzcoiNetal,PRL98,257203(2007)

However,theuniversalityclassisnotfixedtotheexpectedone(k=1WZNWmodel)andcriNcalexponentisconNnuouslychanging

0.5

Wess&Zumino1971,Novikov1981,Wisen1984

2ndPT

Truncation dependence of

The truncation dependence does not converge in the region β ≧ 0.4.

�max

/ L�⌫

1st

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

a/i

`

D`=5,De=3D`=5,De=5

D`=17,De=3

ApplicaNontoCP(1)+θ

TRGdoesnotworkwellnearcriNcalregion⇒TensorNetworkRenormalizaNonshouldbeusedinfuture

Tensornetworkrep.isobtainedthankstocharacter-likeexpansionTRGisusedascoarsegraining

topologicalsuscepNbility

�

�0 � � 1/g2

1stPT

2�

�2ndPT?

0.3

OK Notyetconfirmed

Ple�aetal.1997

CP(1)atθ=π

Kawauchi7/25(Mon)17:05TD

currentstatusofTRGstudy

Evenbly&Vidal2014,Guetal.,2015

MC TRGBoltzmannweightisinterpretedasprobability

Tensornetworkrep.ofparNNonfuncNon(noprobabilityinterpretaNon)

Importancesampling

CompressionoftensorbySVD,VariaNonalapproach

StaNsNcalerrors SystemaNcerrors

Signproblemmayappear

Nosignproblem∵noprobability

CriNcalslowingdown EfficiencyofcompressiongetsworsearoundcriNcality

canbeimprovedbyTNR,Loop-TNRin2DsystemEvenbly&Vidal2014,Guetal.,2015

Summary

n  TensornetworkhasNosignproblemn  Keyofpoint:informaNoncompressionbasedonSVD

n  Various1D/2DsystemsareunderinvesNgaNoninHamiltonian/Lagrangianapproach

n  HigherdimensionalsystemissNllhard…

Futureprospectsn  Longwaytogo4DQCD+μ&θ

n  Cost：O(Dcut15),Memory：O(Dcut

8)for4Dsystem(HOTRG)n  Non-Abeliangaugetheory

Characterexpansion⇒Tensornetworkrep.isOKbutinternald.o.f.ishuge!!!

n  Besercoarsegrainingmethodin4D?(TNR,Loop-TNR)n  EfficientparallelizaNon?

n  Lowdim.systemsufferingfromthesignproblem?n  LaJceSUSY,LaJcechiralgaugetheory,….

n  CombiningwithstochasNcmethodtoreducecost

GoodandBadpoints

n  Freefromthesignproblemn  Largevolumeiseasytodo

Cost∝log(Volume)×(Dcut)const.×Dim.×#DOF

n  Higherdimensionalityishard(2,3isOKbut4ishard)n  Larger#ofinternaldegreeoffreedomisalsohard

Good

Bad

“Good”algorithmcanreduceit

Hierarchyofsingularvalue

•  OffcriNcality:goodhierarchy(smallS)•  NearcriNcality:hierarchygetsworse(largeS)

Tc = 2/[ln(1 +�

2)]= 2.269...

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

10

0 20 40 60 80 100

mm

/m1

m

T=1.70 L=32T=1.70 L=1024T=2.27 L=32T=2.27 L=1024T=2.70 L=32T=2.70 L=1024

likecriNcalslowingdowninMC

S = ��

i

�i ln �i

normalizedsingularvalue

TensornetworkrenormalizaNon(TNR)Evenbly&Vidal2014cancurethesituaNon

Dcut=32

Entanglemententropy

2DIsingmodel

NumericalaspectofTRGandTask

n  matrix-matrixproduct：Level3BLAS　≫　SVDn  BesercoarsegrainingwithsmallDcut(highlycompression)?

n  MaincomputaNon（ForHOTRG,n-dimsystem）n  DecomposiNon　⇒　SVD(EVD)：O(Dcut

6)n  ContracNon　⇒　matrix-matrixproduct：O(Dcut

4n-1)

n  Memoryn  #elementsoftensor：O(Dcut

2n)n  internald.o.f.⇒morememory

Hotspot

n  1Dquantumgappedsystem(e.g.Nsitessystem)n  Target:WavefuncNonn  EfficientwayofchoosingGOODbasisbyusingSchmit

decomposiNon(likeSVD)n  small#ofbasis=informaNoncompressionn  Beforethisappears,limitedtoN=30.ButDMRGenablesN=100n  TensorrenormalizaNongroup(TRG)Levin&Nave2007

n  Target：ParNNonfuncNonofclassicalStat.systemn  ExpressparNNonfuncNonintensornetworkrep.,compresstensorby

usingSVDandcoarsegrainingtensorn  In2Dsystem,verypowerful

Hamiltonianapproach

s1 s2 sN

· · ·

· · · · · · · · ·

DensitymatrixrenormalizaNongroup(DMRG)White1992Schollwoeck2004

ImprovementofCoarsegrainingalgorithmn  TensorEntanglementFilteringRenormalizaNon

n  RemovingshortrangecorrelaNon(parNally)n  worksinoff-criNcalpointbutnotnearcriNcality

n  SecondTRGn  OpNmizaNonincludingenvironment(TRG:locallyopNmal)n  worksinoff-criNcalpointbutnotnearcriNcality

n  TensorNetworkRenormalizaNon(TNR)n  FirstlyremoveshortcorrelaNon(entanglement)byusingdisentangler,andthencoarsegrainingisperformed

n  EvenaroundcriNcality,sustainablecoarsegrainingisrealizedn  LoopTensorNetworkrenormalizaNon(LoopTNR)

n  RobustnessaroundthecriNcalityiscomparablewithTNRn  LesscostO(Dcut

6)whileTNRcostsO(Dcut7)

Guetal.2008

Xieetal.,2009

Evenbly&Vidal2014

Guetal.,2015

ApplicaNontoCP(N-1)+θ

�

�0

ConNnuumlimit

confinement

deconfinement

� � 1/g2

Ple�aetal.1997Seiberg1984

Schierholtz1994

1stPT

Z(�) =�

Q

ei�QP (Q)

toymodelofQCD

StrongCPproblem

2�

�

Strongcouplinglimit

MCCP(3)

StrongcouplingexpansionO(β10)supportthescenario

ApplicaNontoCP(N-1)+θ

�

�0

ConNnuumlimit

confinement

deconfinement

� � 1/g2

Imachietal.1999Ple�aetal.1999

1stPT

Z(�) =�

Q

ei�QP (Q)

toymodelofQCD

StrongCPproblem

2�

�

MCCP(2)

??

MCCP(3)