Non-abelian symmetries in tensor networks: A quantum symmetry

76 Pages·2012·2.25 MB·English

by  Andreas Weichselbaum

AnnalsofPhysics327(2012)2972–3047
ContentslistsavailableatSciVerseScienceDirect
Annals of Physics
journalhomepage:www.elsevier.com/locate/aop
Non-abelian symmetries in tensor networks: A quantum
symmetry space approach
AndreasWeichselbaum
DepartmentofPhysics,ArnoldSommerfeldCenterforTheoreticalPhysics,andCenterforNanoScience,
Ludwig-Maximilians-Universität,80333Munich,Germany
a r t i c l e i n f o a b s t r a c t
Articlehistory: A general framework for non-abelian symmetries is presented
Received10March2012 for matrix-product and tensor-network states in the presence
Accepted8July2012 of well-defined orthonormal local as well as effective basis
Availableonline24July2012
sets. The two crucial ingredients, the Clebsch–Gordan algebra
for multiplet spaces as well as the Wigner–Eckart theorem
Keywords:
for operators, are accounted for in a natural, well-organized,
Non-abeliansymmetries
Clebsch–Gordancoefficients and computationally straightforward way. The unifying tensor-
Liealgebra representation for quantum symmetry spaces, dubbed QSpace,
Numericalrenormalizationgroup is particularly suitable to deal with standard renormalization
Densitymatrixrenormalizationgroup group algorithms such as the numerical renormalization group
Tensornetworks (NRG), the density matrix renormalization group (DMRG), or
also more general tensor networks such as the multi-scale
entanglement renormalization ansatz (MERA). In this paper,
the focus is on the application of the non-abelian framework
within the NRG. A detailed analysis is presented for a fully
screenedspin-3/2three-channelAndersonimpuritymodelinthe
presence of conservation of total spin, particle–hole symmetry,
andSU(3)channelsymmetry.Thesamesystemisanalyzedusing
several alternative symmetry scenarios based on combinations
of U(1) , SU(2) , SU(2) , SU(3) , as well as the
charge spin charge channel
enveloping symplectic Sp(6) symmetry. These are compared in
detail, including their respective dramatic gain in numerical
efficiency. In the Appendix, finally, an extensive introduction
to non-abelian symmetries is given for practical applications,
together with simple self-contained numerical procedures to
obtainClebsch–Gordancoefficientsandirreducibleoperatorssets.
The resulting QSpace tensors can deal with any set of abelian
symmetriestogetherwitharbitrarynon-abeliansymmetrieswith
compact,i.e.finite-dimensional,semi-simpleLiealgebras.
©2012ElsevierInc.Allrightsreserved.
E-mailaddress:[email protected].
0003-4916/$–seefrontmatter©2012ElsevierInc.Allrightsreserved.
doi:10.1016/j.aop.2012.07.009
A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047 2973
Contents
1. Introduction.................................................................................................................................................2973
2. MPSimplementationofnon-abeliansymmetries....................................................................................2976
2.1. Generalquantumspacerepresentation(QSpaces)......................................................................2978
2.2. A-tensorsandoperators.................................................................................................................2980
2.3. Multiplicity......................................................................................................................................2981
2.4. Contractions....................................................................................................................................2984
2.5. Scalaroperators..............................................................................................................................2985
2.6. Operatormatrixelements..............................................................................................................2986
2.7. Densitymatrixandbackwardupdate...........................................................................................2987
3. ImplicationsforDMRGandbeyond...........................................................................................................2988
3.1. GeneralizedA-tensorfortensornetworks....................................................................................2989
3.2. Two-sitetreatment.........................................................................................................................2990
3.3. Statespacetruncation....................................................................................................................2991
3.4. Wavefunctionprediction...............................................................................................................2992
4. Correlationfunctions..................................................................................................................................2992
5. Thenumericalrenormalizationgroup.......................................................................................................2994
5.1. Fulldensitymatrix..........................................................................................................................2994
5.2. Model:symmetricthree-channelsystem.....................................................................................2995
6. NRGresults..................................................................................................................................................2997
6.1. Spectralfunctions...........................................................................................................................2999
6.2. Detailedcomparisonofsymmetrysettings..................................................................................3001
7. Summaryandoutlook................................................................................................................................3002
Acknowledgments......................................................................................................................................3003
AppendixA. Non-abeliansymmetries101...........................................................................................3003
A.1. Simpleexample:rotationalsymmetry..........................................................................................3004
A.2. SU(2)spinalgebra..........................................................................................................................3005
A.3. Generatorsandsymmetrylabels...................................................................................................3007
A.4. ExampleSU(N)...............................................................................................................................3011
A.5. Decompositionintoirreduciblerepresentations..........................................................................3013
A.6. Tensorproductspaces....................................................................................................................3014
A.7. IrreducibleoperatorsetsandWigner–Eckarttheorem...............................................................3015
A.8. Severalindependentsymmetries..................................................................................................3017
A.9. Symmetriesinphysicalsystems....................................................................................................3017
A.10. ThesymplecticgroupSp(2m)........................................................................................................3027
AppendixB. Numericalimplementation..............................................................................................3033
B.1. Tensorproductdecompositionofsymmetryspaces....................................................................3033
B.2. Statespaceinitializationandoperatorcompactification.............................................................3035
AppendixC. ExampleQSpaces..............................................................................................................3036
C.1. FermionicsitewithU(1) ⊗SU(2) symmetry..................................................................3037
charge spin
C.2. Fermionicsitesinthepresenceofparticle–holesymmetry........................................................3039
C.3. ThreechannelswithSU(3)channelsymmetry............................................................................3041
References....................................................................................................................................................3046
1. Introduction
Numerical methods for strongly correlated quantum-many-body systems are confronted with
exponentiallylargeHilbertspaces.Withalimitednumberofexactanalyticalsolutionsathandand
with perturbative treatments for low-energy or ground-state physics often insufficient, a certain
systematic treatment with respect the Hilbert space is required. Besides quantum Monte Carlo
approaches,thatexplorequantumsystemsinastochasticway[1],asystematicstatespacedecimation
is provided by renormalization group (RG) techniques such as the density matrix renormalization
group(DMRG)[2]orthenumericalrenormalizationgroup(NRG)[3],bothhighlyefficientforquasi-
one-dimensionalsystems,andsincenon-perturbative,consideredessentiallyexact.
Quantum-many-body Hilbert spaces are built from the direct product of the state spaces of
the participating individual particles. As such particle statistics plays an essential role. While the
2974 A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047
focus of this paper is on fermionic systems, generalizations to spin systems are straightforward.
Thetreatmentofbosonicsystems,ontheotherhand,comeswiththeadditionalhurdlethateven
a single local bosonic degree of freedom already has an infinite state space of its own which
must be truncated for numerical treatment. Nevertheless, assuming that the bosonic state spaces
can be properly categorized in symmetry sectors, the complications deriving from their infinite
dimensionalityareconsideredseparateformtheissuesregardingthedescriptionofpuresymmetries
oftheHamiltonian.Inthecaseoftwo-dimensionalsystemsfinally,moreexotictypesofparticlesexist
thatareneitherfermionsnorbosons,butanyons.Muchattentionhasbeenpaidtotheserecently
withintheframeworkoftensornetworks[4–9].Whilethetreatmentofparticleswithnon-abelian
statisticsisnicelycomplimentarytotheworkpresentedhere,thisshallnotbepursuedanyfurtherin
whatfollows.
MethodssuchastheDMRGortheNRGthen,arebasedonthesamealgebraicstructureofmatrix
productstates(MPSs)[10,11].Initiallyintroducedforone-dimensionalsystemswithMPSowingits
nametothiscase,awiderangeofactivityhasemergedwithinrecentyearstogeneralizeMPSsto
tensor-networks for two- or higher-dimensional systems [8,12–14]. While clearly appealing from
thepointofviewofarealawsforentanglement-entropy[15–17],tensornetworkstates(TNSs)often
sharethesamedisadvantageaslinearsystemswithperiodicboundaryconditionswithintheDMRG,
namelythatstatespacesbecomeintrinsicallynon-orthogonal.Thereforealsotheuniqueassociation
ofsymmetrylabelswitheachindexinatensoriscompromised.This,however,canbecircumvented
byintroducinganemergingextra-dimension,whichisatthebasisoftherecentlydevelopedmulti-
scale entanglement renormalization ansatz (MERA) [18,19]. Nevertheless, the traditional DMRG
approach applied to 2D systems [20] with open or cylindrical boundary conditions yet with long-
rangeinteractionshascontinuedtoprovideahighlycompetitive,extremelywell-controlled,even
thoughnumericallyexpensiveapproach.
Withinboth,traditionalDMRGaswellasNRG,statespacesofentireblocksarebuiltiteratively
byaddingandmergingonesiteatatime.Clearly,thesingleindexdescribinganeffectivebasisfor
theentireblockorsitecanbechosenorthogonal.Inaddition,basisstatescanbelabeledinterms
ofthesymmetriesoftheunderlyingHamiltonian.Operatorswrittenasmatrixelementsinthisvery
same basis therefore also share the same well-defined partitioning in terms of symmetry sectors.
Bygroupingsymmetrystatespacestogether,theHamiltonianbecomesblock-diagonal,whilemore
generaloperatorsusuallyobeywell-definedselectionrulesbetweensymmetrysectors.Consequently,
the sparsity of these operators due to symmetry can be efficiently and exactly included in the
numericaldescription,suchthatusuallyonlyafewdensedatablockswithnon-zeromatrixelements
remain,giventhesymmetryconstraints.Whilethiswellrepresentstheadvantageofimplementing
generic abelian symmetries in a calculation, the presence of non-abelian symmetries offers yet
anotherstrongsimplification:manyofthenon-zeromatrixelementsareactuallynotindependentof
eachother,bearinginmind,forexample,theWigner–Eckarttheorem.Thereforegoingbeyondabelian
symmetries,non-abeliansymmetriesallowtosignificantlycompressthenon-zeroblocksintermsof
multipletspaces[21–23].Inparticular,thisremovesthemixingofsymmetrysectors,hadonlyabelian
symmetriesbeused,instead.WiththeClebsch–Gordancoefficientspacesfactorizing[8,9,24],theycan
besplitoffsystematicallyintermsofatensor-productanddealtwithseparately.
AnMPSisoptimalforone-dimensionalsystems.Whenexploringsystemsthatarenotstrictlyone
dimensionalbutacquirewidth,suchasladdersofseveralrungsinDMRGormulti-channelmodels
in NRG, the price to be paid for orthonormal state spaces is that one must represent the system
as a one-dimensional MPS nevertheless. This introduces longer-range interactions to the mapped
1D system, with the effect that the typically required dimensions of the state spaces to be kept
in a calculation, grow roughly exponentially with system width. The number of symmetries then
that(i)areavailableand(ii)arealsobeexploitedinpractice,decideswhetherornotacalculation
is feasible. Abelian symmetries such as particle (charge Q) or spin (S ) conservation are usually
z
implementedinDMRGcalculations.However,onlyveryfewgroupshaveimplementednon-abelian
( )
symmetries,andthesearealsoconstrainedtoSU 2 symmetriesonly[21,22],duetoitscomplexity
intheactualimplementation.Generaltreatmentofnon-abeliansymmetrieswithintheMERA,onthe
otherhand,iscurrentlyunderdevelopment[8,25].NRG,incontrast,hadbeensetupincludingnon-
( )
abelianSU 2 spinsymmetryfromitsverybeginning[3],dictatedbylimitednumericalresources.So
A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047 2975
far,however,onlyaveryfewisolatedattemptsincludingmorecomplexnon-abeliansettingsexist
withintheNRG[26],whiletoourknowledgethereexistsnogeneralrealizationyetofarbitrarynon-
abeliansymmetriesineithermethod.
Thispaperfocusesonthesystematicdescriptionandimplementationofnon-abeliansymmetries
of a given Hamiltonian within the generalized MPS framework through explicit evaluation to
Clebsch–Gordan coefficients. This naturally also includes the description of abelian symmetries,
as they can be trivially written in terms of Clebsch–Gordan coefficients. While the focus within
( ) ( )
non-abelian symmetries belongs to SU N and the symplectic group Sp 2n , the generalization
to other non-abelian symmetries or also point groups is straightforward once their particular
( )
Clebsch–Gordancoefficientsareworkedout.Incontrasttothewell-knownSU 2 then,generalnon-
abeliansymmetries,suchasSU(N ≥ 3),representasignificantincreaseinalgorithmiccomplexity,
inthattheycanandroutinelydoexhibitinnerandoutermultiplicity.Thelatter,forexample,implies
thatinthedecompositionofthetensor-productoftwoirreduciblerepresentations(IREPs)intoadirect
sumofIREPs,thesameIREPmayoccurmultipletimes.Nevertheless,thiscanbedealtwithproperly
onthealgorithmiclevel,aswillbeshownindetailinthispaper.
Whilethepresentednon-abelianframeworkforgeneraltensorsisstraightforwardlyapplicableto
traditionalDMRGaswellasNRG,thepaperfocusesontheapplicationwithintheNRG.Detailedresults
( )
arepresentedforafullyscreenedspin-3/2AndersonimpuritymodelwithSU 3 channel-symmetry
[i.e.seeHamiltonianinEq.(27)].ThismodelhasbeensuggestedastheeffectivemicroscopicKondo
model for iron impurities in gold or silver [27], historically the first system where Kondo physics
was observed experimentally [28,29]. Being a true three-channel system, this cannot be trivially
rotatedintoasimplerconfigurationoffewerrelevantchannels.Theresultisanextremelychallenging
calculation within the NRG that requires non-abelian symmetries for fully converged numerical
resultsforreasonablecoarse-grainingofthecontinuousbath.Thenon-abeliansymmetriespresentin
themodelconsideredare(i)particle–holesymmetryineachofthethreechannels,SU(2)⊗3 ,(ii)total
charge
( ) ( )
spin symmetry, SU 2 , and (iii) channel symmetry, SU 3 . The non-abelian particle–hole
spin channel
( ) ( )
SU 2 symmetry, however, does not commute with the channel SU 3 symmetry, while the plain
abelianchargeU(1)symmetrydoescommute.Overall,thissuggestsalargerenvelopingsymmetry,
( ) ( )
which turns out to be the symplectic symmetry Sp 6 [30] [for an introduction to Sp 2n , see
AppendixA.10].Withthis,thefollowingsymmetryscenariosareconsideredandcomparedindetail,
SU(2) ⊗SU(2)⊗3 ,
spin charge
SU(2) ⊗U(1) ⊗SU(3) , and
spin charge channel
SU(2) ⊗Sp(6).
spin
( )
While the first setting represents a more traditional setup based on multiple sets of plain SU 2
( )
symmetries only, the second setting already includes the larger channel SU 3 symmetry. Both of
thesesymmetriesdonotcapturethefullsymmetryofthemodel,whichfinallyisachievedbyusing
( )
theenvelopingSp 6 symmetry.
( )
Due to the internal two-dimensional structure of the SU 3 symmetry based on the fact that
( )
SU 3 hastwocommutinggenerators,i.e.isofrank2,itsmultipletshavesignificantlylargerinternal
dimension, in practice, up to over a hundred. Therefore despite the reduction of the particle–hole
( )
symmetrytoaplainabeliansymmetry,thesecondsettingwiththeSU 3 channelsymmetryallows
( ) ( )
tooutperformthemoretraditionalsetupbasedonSU 2 symmetriesonly.Similarly,withSp 6 a
rank-3symmetry,multipletstheneasilyreachdimensionsofseveralthousandsthere,whichallows
toreducemultipletspacessignificantlyfurtherstill.Adetailedanalysisofthisisprovidedinthispaper,
withamoregeneralself-containedintroductiontonon-abeliansymmetriesconsideredgiveninthe
Appendix[cf.AppendixC.3].
FromanNRGpointofview[31],afewessentialstepsarerequired.Theseare(i)theevaluationof
relevantoperatormatrixelementsrequiredtoconstructtheHamiltonian,(ii)thegenericsetupofan
iteration,addingonesitetotheso-calledWilsonchain,andfinally,forthermodynamicalproperties
(iii)alsothetreatmentofthefullthermaldensitymatrix[31].Allofthesestepsaresimpleinprinciple,
yet come with the essential challenge to have a flexible transparent framework for the treatment
2976 A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047
of non-abelian symmetries in practice. In this paper, such a framework is presented in terms of
generalizedcontractionsoftensorsinthepresenceofsymmetryspaces,introducedasQSpacesbelow.
Thepaperisthusorganizedasfollows.Section2describestheMPSimplementationofnon-abelian
symmetries in terms of QSpaces. Section 3 points out specific implications for DMRG and more
general tensor networks. Section 4 describes the implications for calculating correlation functions
in the presence of irreducible operator sets. Section 5 gives a short review of the NRG together
with specialties related to non-abelian symmetries, such as calculating reduced density matrices.
ThissectionalsointroducesthemodelHamiltonianofafullysymmetric3-channelAndersonmodel.
Section 6 then presents explicit NRG results, followed by summary and outlook. Finally, also an
extendedAppendixhasbeenaddedtothepaper.Thelatterisintendedtoprovideamoregeneral
pedagogicalself-containedintroductiontonon-abeliansymmetriesastheyoccurinfermioniclattice
models,togetherwiththeiractualimplementationinpracticeintermsofQSpaces.
2. MPSimplementationofnon-abeliansymmetries
ˆ
ConsidersomeHamiltonianHthatisinvariantunderasetofn symmetries,
S
S ≡nS Sλ, (1)
λ=1
thatis,[Hˆ,Sˆαλ]=0,whereαidentifiesthegeneratorSˆαλforthesimple(non-abelian)symmetrySλ.To
bespecific,forexample,S =SU(2) ⊗SU(2) ≡S1⊗S2withλ∈{1,2}wouldstandforthe
spin charge
( )
combinationofspinandchargeSU 2 symmetry,respectively.Thetensor-productnotationinEq.(1)
indicatesthatthesymmetriesactindependentlyofeachother,thatis[Sˆαλ,Sˆαλ′′]=0forλ̸=λ′.
ˆ
GiventhesymmetriesasinEq.(1),thisallowstoorganizethecompletebasisofeigenstatesofH
intermsofthesymmetryeigenbasis.Everystatethenbelongstoawell-definedirreduciblemultiplet
λ λ
q foreachsymmetryS .Themultipletitselfhasaninternalstatespacestructurethatisdescribedby
theadditionalquantumlabelsqλ.Forexample,inthecaseofSλ =SU(2),qλ(qλ)correspondstothe
z z
spinmultipletS(theS label),respectively.
z
Thusallstatesinagivenvectorspacecanbecategorizedusingthehierarchicallabelstructure
|qn;q ⟩ (state-spacelabelstructure), (2)
z
where
(i) q ≡ (q1,q2,...,qnS),tobereferredtoasq-labels(quantumlabels),referencestheirreducible
representations(IREPs)foreachsymmetrySλ,λ = 1,...,n .AllstatesingivenHilbertspace
S
withthesameq-labelsareblockedtogether,tobereferredtoassymmetryblockq.
(ii) Givenasymmetryblockqthen,themultipletindexn(q)identifiesaspecificmultipletwithinthis
space.Itisthereforeaplainindexassociatedwithgivensymmetryspaceq.Togetherwiththeq-
labels,thisformsthemultipletlevelwhichisconsideredthetopmostconceptuallevel.Usingthe
( )
compositenotation qn toidentifyanarbitrarymultiplet,thesubscriptqtothemultipletindex
n(q)isconsideredimplicitandhenceisdropped,forsimplicity.
(iii) Finally,thesetoflabelsq ≡ (q1,q2,...,qnS),tobereferredtoasz-labels,resolvestheinternal
z z z z
λ λ
structure of each multiplet in q. That is, for each IREP q , referring to the symmetry S in q,
λ
q labels its internal IREP space. As such, the z-labels are entirely defined by the symmetries
z
ˆ
considered.Byconstruction,theeigenstatesoftheHamiltonianHarefullydegenerateinthez-
labels.
Herethesymmetrylabelsqandq describethecombinedrecordoflabelsderivedfromallsymmetries
z
considered.Inpractice,statescanmostlybetreatedonthehighermultipletlevel,whilethelowerlevel
intermsofthez-labelsissplitoffandtakencareofbyClebsch–Gordanalgebraandthecoefficient
spacesderivedfromit.
Whennon-abeliansymmetriesarebroken,theyareoftenreducedtotheirabeliansubalgebra.This
canbeeasilyimplemented,nevertheless,consistentwiththepresentedframework.Inparticular,in
A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047 2977
theabeliancase,thenon-abelianmultipletlabelsqareabsent,whiletheabelianq quantumnumbers
z
remain.Thereforetheq labelscanbepromotedtothestatusofq-labels,q := q .Asaconsequence,
z z
theconceptoftheactualq labelsbecomesirrelevant(thereforesubsequently,theq labelspacemay
z z
simplybesettozero,q :=0).ThecorrespondingClebsch–Gordancoefficientsarealltrivialscalars,i.e.
z
equalto1.Yetthese‘‘Clebsch–Gordancoefficientsforabeliansymmetries’’domaintainanimportant
role,inthattheytakecareoftheproperadditionrulesthatcomewithabeliansymmetries,resulting
in⟨q1;q2|q⟩=1·δq,q1+q2.
GiventheMPSbackgroundofNRGorDMRG,statesspacesaregeneratediteratively,intermsofa
product-spaceofagiveneffectivestatespacewithanewlyaddedlocalsite.Operators,ontheother
hand,aretypicallyrepresentedinlocalstatespaces,andstartingfromthere,theycanbewrittenin
termsofmatrixelementsintheeffectiveglobalstatespaces.Withthisinmind,theimplementation
ofnon-abeliansymmetrieswithintheMPSframeworkthereforeisbasedonthefollowingtwobasic
observationswithrespecttostatespaceandoperatorrepresentations,respectively.
(1) Statespaces:considertwodistinctstatespaces,|Qn;Q ⟩and|ql;q ⟩that,forexample,represent
z z
alargeeffectivestatespaceandasmallnewlocalstatespace,respectively.Assumingthatboth
statespacesallwell-categorizedintermsofIREPs,thentheirtensor-productspacecanalsobe
decomposedintoadirectsumofnewcombinedIREPs|Q˜n˜;Q˜ ⟩usingClebsch–Gordancoefficients
z
(CGCs),
|Q˜n˜;Q˜ ⟩=  A[q][l] ·C[qz] |Qn;Q ⟩|ql;q ⟩. (3)
z QQ˜ nn˜ QzQ˜z z z
Qn;Qzql;qz
NotethattheClebsch–GordancoefficientsgivenbyC[qz] ≡ ⟨QQ ;qq |Q˜Q˜ ⟩(i)fullydefinethe
QzQ˜z z z z
internalmultipletspaceasspecifiedbytheLiealgebra,and(ii)determinethesplitting,i.e.which
˜
output multiplets Q occur for given multiplets Q and q. On the multiplet level, on the other
hand,where(A[q])[l] combinesthemultipletsQnandqlintothemultipletQ˜n˜ consistentwith
QQ˜ nn˜
the splitting provided by the CGCs, the coefficients (A[q])[l] may encode an arbitrary unitary
QQ˜ nn˜
transformationwithinthen˜ outputspaceforeachQ˜.Ther.h.s.ofEq.(3)demonstrates,thatthe
CGCspacesclearlyfactorizefromthemultipletspaceA[q] asatensorproduct.
QQ˜
(2) Operators:withinsomestatespace|Qn;Q ⟩,thematrixelementsofanirreducibleoperator(IROP)
z
set Fˆq [i.e. a set of operators that transforms according to multiplets q for given symmetries;
cf.Eq.(A.3b),oralsoAppendixA.7]canbewrittenusingtheWigner–Eckarttheoremas[32]
⟨Q′n′;Qz′|Fˆqqz|Qn;Qz⟩=FQ[qQ]′[n1n]′ ·CQ[qzzQ]z′, (4)
withC[qz]′ againtheClebsch–GordancoefficientsasinEq.(3).Onthemultipletlevel,thereduced
QzQz
matrixelements(FQ[qQ]′)[n1n]′ ≡⟨Q′n′∥Fˆq∥Qn⟩refertothesingleirreducibleoperatorsetlabeledbyq,
whichisindicatedbythesuperscript[1].TheWigner–Eckarttheoremthusallowstocompactify
the operator matrix elements on the l.h.s. of Eq. (4) as the tensor-product of reduced matrix
elementsandCGCs,asshownonther.h.s.ofEq.(4).
Therefore in both cases above, i.e. in all tensor objects relevant for a numerical calculation, the CGC
spacesfactorize.Thisallowstostronglycompresstheirsize,andthustodrasticallyimproveonoverall
numericalperformance.Moreover,notethatinbothcases,Eq.(3)aswellasEq.(4)theunderlying
structurecomprisestensorsofrank-3throughout.Thisrank-3structureholdsforboth,thereduced
multipletspaceaswellastheCGCspaces.Therefore,ineithercase,thefinaldatastructureofeither
statespacedecompositionaswellasreducedoperatorsetsisexactlythesame.Itisimplemented,in
practice,intermsofwhatwillbereferredtoasQSpaceforgeneraltensorsofarbitraryrank.
2978 A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047
2.1. Generalquantumspacerepresentation(QSpaces)
Thegenericrepresentation,usedinpracticetodescribeallsymmetryrelatedtensorsB,isgivenby
alistingofthefollowingtype,
 q-labels reducedspace∥B∥ CGCspaces 
B≡ {Q}1 B1 {C}1 . (5)
 {·Q·}·2 ·B·2· {·C·}·2 
ˆ
Bynotationalconvention,anactualoperatorBwillbewrittenwithahat,whileitsrepresentationin
termsofmatrixelementsinaspecificbasiswillbewrittenwithoutthehat,hencethecorresponding
QSpaceisreferredtoasQSpaceB.ManyexplicitexamplesofQSpacesareintroducedanddiscussed
indetailinAppendixC.Asanup-frontillustration,consider,forexample,thegeneralHamiltonianof
( )
asinglespinfulfermionicsiteinthepresenceofSU 2 symmetryinthespin(S)andchargesector(C),
whichcanbewrittenastheQSpace[seeEq.(C.8)]
 (S;C) (S′;C′) ∥H∥ CGCspaces 
H ≡ 12;0 12;0 h1,0 1(2) 1. . (6)
 0;12 0;21 h02,1 1. 1(2) 
2
Witheverynon-zeroblocklistedasanindividualrow,onecanseethattheonlytworeducedmatrix
elements∥H∥freetochoosewithoutcompromisingtheSU(2)⊗2 ≡SU(2) ⊗SU(2) symmetry
SC spin charge
aretheparameters(numbers)h1/2,0 andh0,1/2.Bydefinition,theHamiltonianisascalaroperator,
thereforeitistheonlyoperatorwithinitsIROP,hencecanbewrittenasplainrank-2QSpace(the
thirddimensionforthisIROPwouldbeasingletondimension,hencecanbedropped).Beingascalar
operator,theHamiltonianisblockdiagonal,whichisreflectedinequalsymmetrysectors(S;C)and
(S′;C′)ineachrowforthefirstandtheseconddimension,respectively.Moreover,ingivencase,the
correspondingClebsch–Gordancoefficient(CGC)spacesalsoresultintrivialidentities,with1(2)the
two-dimensionalidentity.NotethatthefullsetofCGCspacesineachrowneedstobeinterpretedas
appearinginatensorproductwiththemultipletspace,herethereducedmatrixelementsh1/2,0 or
h0,1/2[e.g.seeEq.(8)below].
Ingeneral,therepresentationofatensorBofarbitraryrank-rintheQSpaceinEq.(5)[withEq.(6)
anexampleforarank-2QSpace],onlyliststhenon-zero,i.e.relevantsymmetrycombinations.Having
r tensordimensions,eachofitsr indicesreferstoitsspecificstatespace|qn;q ⟩ ≡ |(q)n;(q )⟩
z i i i z i
withi = 1,...,r,andhencecarriesitsownlabelstructureasinEq.(2).Theq-labels(q) ≡ {qλ}
i i
λ
alreadyrepresentthecombinedsetofn IREPlabelsfromallsymmetriesS forthestatespaceat
S
( ) λ
tensordimensioni.Ingeneral,byconvention,theinternalorderoftheq-labels q w.r.t. isfixed
i
andfollowstheorderofsymmetriesusedinEq.(1).
ν
Foracertainrow oftheQSpacelistinginEq.(5)then,thesetofrq-labelsaregroupedinto
{Q}ν ≡{(q)1,...,(q)r}ν. (7a)
The reduced matrix elements are stored in the dense rank-r tensor Bν indexed by ni with i =
,...,
1 r.Thisisaplaintensor,withthemultipletspacespossiblyalreadyrotatedbyarbitraryunitary
transformationsandtruncated.Thisisalsoreflectedinthefactthattheindicesn areplainindices,i.e.
i
carrynofurtherinternalstructure.Finally,foreveryoneoftheλ=1...n symmetriesincluded,the
S
correspondingCGCspaceisstoredinthesparsetensorsCλ,ν,eachofwhichisalsoofrankr.TheseCGC
spacesaregroupedinto{C}νinthelastcolumn,
{C}ν ≡{C1;ν,...,CnS;ν}. (7b)
Astheq-labels{Q}ν alsodefinethez-labels,thereisnoexplicitneedtostorethez-labels(qz)i,ν.The
internal running indices in Cλ,ν, however, are uniquely associated with the z-labels. Note also the
differentindexsetting:incontrasttoEq.(7a),whichcontainsasetofr q-labels,i.e.oneforevery
A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047 2979
dimensionoftherank-r tensorB,Eq.(7b)containsasetofn rank-r CGCspaces,i.e.oneforevery
S
symmetry.
InadditiontotheQSpacelistinginEq.(5),alsothetypeandorderofsymmetriesconsideredis
storedwithaQSpace,cf.Eq.(1),eventhoughthisisusuallythesamethroughoutanentirecalculation.
ν
Moreover,notethattheroworrecordindex inEq.(5)ispurelyforconveniencewithoutanyspecific
meaning,astheorderofrecordsinaQSpacecanbechosenarbitrarily.Nevertheless,itisrequiredto
refertoaspecificentryinaQSpace.
ν
Foragivenrecord intheQSpaceinEq.(5)then,thereducedspaceandtheCGCspacesaretobe
interpretedasanoveralltensor-product,
 
nS
Bν ⊗{C}ν ≡Bν ⊗ Cs;ν , (8)
s=1
which,ofcourse,isneverexplicitlyreexpandedintoasinglefullobjectatanystep,inpractice.Eq.(8)
ratherdemonstratesthesinglemostimportantmotivationtoimplementnon-abeliansymmetriesina
numericalcomputation.BysplittingofftheCGCspacesintermsofatensorproduct,blockdimensions
can be strongly reduced for larger calculations with several symmetries present. For the models
analyzedinthispaper,forexample,thiswastypicallyanaveragedimensionalreductionfromplain
abeliansymmetriesbyafactorof10uptoseveralhundreds.ConsideringthatbothNRGandDMRG
scalelikeO(D3)withDthetypicaldimensionofdatablocks,thisisanenormousgaininefficiency.
ThefactorizedCGCspaces,ontheotherhand,canbedealtwithindependently,aswillbeexplained
indetaillater.AssumingthatusuallythedimensionsofthereducedstatesspacesBν stillexceedby
farthetypicaldimensionsencounteredfortheCGCspaces,thelatterbearlittlenumericaloverhead.
( )
Onlyforlarger-ranksymmetries,suchasthesymmetrySp 6 discussedlater,multipletdimensions
canbecomelargethemselvessuchthatoneneedstopaymoreattentiontoanefficienttreatmentof
theircorrespondingCGCspaces[seeAppendixC.3.2].
ForQSpaceswheretheCGCspacesinEq.(5)exactlycorrespondtothestandardClebsch–Gordan
coefficientsforeachsymmetry,onemayarguesimilartothez-labels,thatactuallyitisnotexplicitly
necessary to store the CGC spaces together with the QSpace, since these are known. This is true,
indeed,fortheseparticularcases,andCGCspacesmaysimplybereferencedthen.Nevertheless,the
explicitstorageoftheCGCspaceswithaQSpaceasinEq.(5)haspracticalvalue.Whencombining
QSpacesthroughcontractions,i.e.sumoversharedindices,forexample,quitefrequentlyintermediate
objectscanarisethatdohaverankdifferent,inparticularalsolargerthan3[e.g.seetheintermediate
objects X indicated by the dashed boxes in Fig. 4]. These then elude a description in terms of
standard rank-3 CGCs. In this case, the actual CGC spaces for intermediate QSpace are important,
andeventhoughtheydonotnecessarilyresembletheinterpretationoftheoriginalstandardrank-3
spacesofstandardClebsch–Gordanalgebraanymore,thesespaceswillbereferredtoasCGCspaces
nevertheless,owingtotheirorigin.
Furthermore,forspecificalgorithmssuchasNRGandDMRG,onagloballevelonetypicallydeals
withsimplescalaroperatorssuchastheHamiltonianoradensitymatrix,apartfromintermediate
steps where complex CGC structures can arise. Therefore the full sequence of contractions on the
CGClevel[e.g.seeFig.4]canbereplacedbyanalyticalexpressionsorsumrulesforClebsch–Gordan
coefficients.Inparticular,inmanysituationstheexplicitknowledgeof3j-and6j-symbols,ormore
( )
general 3n -jsymbols,appearssufficient[7,22,24,33,34]withcurrentapplicationsinthisdirection
( ) ( )
again mainly restricted at most to SU 2 . If the 3n -j symbols were known for arbitrary non-
abelian symmetry, the explicit storage of the CGC spaces with the QSpaces would no longer be
( )
required,indeed,andcouldbeavoidedaltogether.Note,however,that 3n -jsymbolsrequirespecific
contractions which must be implemented within the code dependent on the context. While vast
( )
literature exists on 3n -j symbols, this is limited to an overwhelming extent on the relatively
( )
simple symmetry of SU 2 , for which analytic expressions exist, indeed. For arbitrary non-abelian
( )
symmetries,however,the 3n -jsymbolsmayormaynotbeknown[35].FortheQSpaceasoutlined
inthispaper,ontheotherhand,nospecialtreatmentisrequiredforspecificcontractions,andno
explicitknowledgeofpossiblysymmetrydependentCGCsumrulesisrequired.TheQSpaceapproach
solelyreliesonthecorrectconstructionofthestandardCGCspacestostartwith,withthesubsequent
2980 A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047
sumsoverCGCspacesperformedexplicitlynumericallyandnotanalyticallythroughexactlythesame
contractionasonthereducedmultipletlevel,asdiscussedinmoredetaillater.
Finally,theexplicitinclusionoftheCGCspacesallowstobuildinstrongconsistencychecksinthe
actualnumericalimplementation.ImaginethattheHamiltonianisbuiltbyasequenceofcomplex
contractions. The Hamiltonian eventually must be a scalar operator, i.e. it is block diagonal in the
symmetriesandtheCGCspacesreducetoplainidentities.Thiscansimplybecheckedattheendofthe
calculation,whichthusprovidesastrongcheckofwhetherthesymmetrieshavebeenimplemented
correctly or not. At the stage of intermediate contraction, however, the CGC spaces guarantee the
correctsplittingandweightdistributionbetweendifferentemergingsymmetrysectors.
2.2. A-tensorsandoperators
ConsidertheprototypicalMPSscenarioasinEq.(3)thattakessomepreviouslyconstructedstate
space|i⟩ ≡ |Qn;Q ⟩andaddsanewlocalstatespace|σ⟩ ≡ |ql;q ⟩,e.g.anewphysicalsite.The
z z
statespacesarethuscombinedinaproduct-spacedescribedintermsoftheIREPs|j⟩ ≡ |Q˜n˜;Q˜ ⟩.
z
σ
Herethestatesi, ,andjareintroducedasnotationalshorthandforbetterreadability.Theproduct
spacethenisspannedby|iσ⟩≡|σ⟩|i⟩.Theorderofstatesinthelatterproductemphasizesthatstate
|σ⟩istypicallyaddedafter andthusontotheexistingstate|i⟩,whichisofparticularimportancefor
fermionicsystems.Ingeneral,thecombinedstates[36,37]

|j⟩= |iσ⟩⟨iσ|j⟩, (9)
lσ ≡A[σ]
ij
are described in terms of linear superpositions of the product space |iσ⟩ given by the coefficients
[σ] [σ] [σ]
A ,henceforthcalledA-tensor(rank-3)orA -matrices(rank-2).Withouttruncation,A denotesa
ij σ ij
fullunitarymatrixU(iσ),jwheretheroundbracketindicatesthattheindicesiand havebeenfused,
i.e.combinedintoaneffectivesingleindex.Thepresenceofsymmetryandthepropercategorization
ofstatespaces,however,imposescertainconstraintsonthisunitarymatrix,aspointedoutalready
withEq.(3).Inparticular,thefullydeterminedCGCspacesC[qz] factorizefromtheA-tensor,allowing
QzQ˜z
an arbitrary rotation in the reduced multiplet space A[q] only. For the specific case then, that the
QQ˜
reduced multiplet spaces are identical to partitions of identity matrices with a clear one-to-one
correspondencestillofinputandoutputmultiplets,thecorrespondingA-tensorwillbereferredto
astheidentityA-tensor[seeFig.2later;forexplicitexamples,seeEq.(C.4)orEq.(C.6)].AnidentityA-
tensorthereforerepresentsthefullstatespacestillwithoutanystatespacetruncation,andisunique
uptopermutationsinthecombinedoutputspace.Itsexplicitconstructionisaconvenientstarting
point,inpractice,whenmergingnewlocalstatespaceswithexistingeffectivestatespaces.
TheentireconstructionofanA-tensorcanbeencodedcompactlyintermsofarank-3QSpace.Both
coefficientspacesinEq.(3),C[qz] aswellasA[q],directlyentertheQSpacedescriptioninEq.(5).A
QzQ˜z QQ˜
schematicpictorialrepresentationofanA-tensorisgiveninFig.1.Therethestatesi(j)representthe
σ
opencompositeindextotheleft(right),respectively,while referstotheopencompositeindexat
thebottom.
AsalreadyarguedwithEq.(4),anirreducibleoperatorsharesexactlythesameunderlyingCGC
structureasanA-tensor.ThusalsoitsrepresentationintermsofaQSpaceiscompletelyanalogous.
ConsideranIROPsetFˆq ≡ {Fˆq},whichtransformsaccordingtoIREPq.Here,thecompositeindex
σ ≡ (ql;q ), for short, identqizfies the specific operators in the IROP set. As already indicated by
z
thesuperscript[1]inEq.(4),itsassociatedmultipletindexlhasthetrivialrangel = 1,since,by
definition,theIROPrepresentsasingleIREPontheoperatorlevel.Withthestates|i⟩≡|Q′n′;Q′⟩and
z
|j⟩ ≡ |Qn;Q ⟩nowrepresentingthesamestatespacewithinwhichtheoperatoracts,withusually
z
manymultipletsanddifferentsymmetries,theoperatorrepresentationoftheIROPFˆq inthestates
⟨i|and|j⟩isevaluatedusingtheWigner–EckarttheoreminEq.(4).SimilartotheA-tensorearlier,the
resultingfactorizationoftheCGCspacesCQ[qzzQ]z′ togetherwiththeremainingmultipletspaceFQ[qQ]′ of
reducedmatrixelementsdirectlyentertheQSpacedescriptioninEq.(5).
A.Weichselbaum/AnnalsofPhysics327(2012)2972–3047 2981
Fig.1. (Coloronline)Schematicdepictionofarank-3QSpaceasanexampleforabasicbuildingblockforanMPSoratensor
network,wherelines(boxes)representindices(dataspaces),respectively.Everyindexisassumedtorefertoastatespace
withsimilarphysicalbackground,hencereferstothesameglobalsymmetriesasinEq.(1),andhasthegenericcomposite
structure|qn;qz⟩asinEq.(2),whereqz specifiesthestateswithintheCGCspaces.Therank-3QSpacedepictedcanbe
interpretedintwoentirelydifferentwayswhilesharingexactlythesameunderlyingalgebraicstructure.Theseare(i)the
statespacedecompositionintoIREPsand(ii)operatorrepresentationforagivenIROPinagivenbasis(seetext).Forthegeneral
interpretationoftheQSpacedepicted,considerforsimplicityasinglerowνinEq.(5).Theset{Q}νdefinestheq-labelsfor
alltensordimensions(hereatotalofthree).Forgivenq-labelsthen,thecorrespondingmultipletindicesnkeeptrackofthe
reducedmultipletspaces.Thesearetypicallylargestill,asindicatedbythethickblacklinesforeachtensordimension.The
correspondingreducedrank-3multipletspaceAνisdepictedbythelargegrayboxinthebackground.Withtheq-labelsfixed,
thisalsospecifiestheIREPsforeverytensordimensionandeverysymmetry.ThecorrespondingsparseCGCspacesareindicated
bythesmallboxesaroundthecenter,withoneboxforeverysymmetry.Examplesareabelianparticleconservation,non-abelian
spinSU(2),non-abelianchannelSU(3),orother.Byconstruction,allCGCspacessharethesamerankastheunderlyingQSpace.
ThereforeeachCGCspacealsohasthreelinesattached,oneforeverytensordimension.Fornon-abeliansymmetries,ingeneral
theCGCspacesrefertofinitemultipletdimensions,whileforsimplersymmetries,suchasabeliansymmetries,theCGCspaces
actuallybecometrivial,i.e.scalars.These,nevertheless,arealsointerpretedashavingthesamerankastheQSpacebyusing
singletondimensionswhereverrequired.
Soeventhoughanoperatorisusuallyconsideredarank-2object,thefactthatanIROPconsistsof
σ
anoperatorsetindexedby ,addsathirdindextotheQSpace.Incontrasttothestateinterpretationof
σ σ
fortheA-tensorabove,however,herethe‘‘index’’ hasadifferentinterpretationinthatitpoints
σ
toaspecificoperatorintheIROPset.Byconvention,theoperatorindex willalwaysbelistedas
thirdtensordimensioninitsQSpacerepresentation.Giventhethree-dimensionalrepresentationof
ageneralIROP,thereforeitsentireconstructionmimicstheconstructionofanA-tensorintermsofa
QSpace.Asaconsequence,Fig.1exactlyalsoresemblestheQSpacestructureofanIROP.Thestatesi(j)
usedforthecalculationofthematrixelementrepresenttheopenindextotheleft(right),respectively,
σ
whiletheoperatorindex referstotheopenindexatthebottom.
Scalaroperators,finally,suchastheHamiltonianofthesystemordensitymatrices,representa
specialcase,sincetheretheIROPsetcontainsjustasingleoperator.Thereforethethirdindex,i.e.the
operatorindex,becomesasingletonandhencecansimplybedropped[e.g.seeEq.(6)].Scalaroperators
thereforearerepresentedbyrank-2QSpaces.Theyareblock-diagonalintheirsymmetries,andtheir
CGCspacesareallequaltoidentitymatrices,withanexamplealreadygiveninEq.(6).
2.3. Multiplicity
Forgeneralnon-abeliansymmetries,frequentlyinnerandoutermultiplicityoccur[38,39].Both
areabsentinSU(2),yetdooccuronaregularbasisforSU(N ≥ 3).Innermultiplicitydescribesthe
situationwhereforagivenIREP,severalstatesmayshareexactlythesamez-labels.Letmq denote
z
thenumberoftimesaspecificz-labeloccurswithinIREPq.Thenthepresenceofinnermultiplicity
implies mq > 1 for at least one z-label. Within such degenerate subspaces an arbitrary rotation
z