REVIEWLETTERS126,120604(2021)
QuasisymmetryGroupsandMany-BodyScarDynamics
JieRen,1,
2,*ChenguangLiang,1,
2,*andChenFang1,3,
4,† 1BeijingNationalLaboratoryforCondensedMatterPhysicsandInstituteofPhysics,ChineseAcademyofSciences,Beijing100190,China 2UniversityofChineseAcademyofSciences,Beijing100049,China3SongshanLakeMaterialsLaboratory,Dongguan,Guangdong523808,China4KavliInstituteforTheoreticalSciences,ChineseAcademyofSciences,Beijing100190,China (Received29July2020;revised6December2020;epted24February2021;published24March2021) Inquantumsystems,asubspacespannedbydegenerateeigenvectorsoftheHamiltonianmayhavehighersymmetriesthanthoseoftheHamiltonianitself.Whenthisenhanced-symmetrygroupcanbegeneratedfromlocaloperators,wecallitaquasisymmetrygroup.WhenthegroupisaLiegroup,anexternalfieldcoupledtocertaingeneratorsofthequasisymmetrygroupliftsthedegeneracy,andresultsinexactlyperiodicdynamicswithinthedegeneratesubspace,namely,themany-body-scardynamics(giventhatHamiltonianisnonintegrable).Weprovidetworelatedschemesforconstructingone-dimensionalspinmodelshavingon-demandquasisymmetrygroups,withexactperiodicevolutionofaprechosenproductormatrix-productstateunderexternalfields. DOI:10.1103/PhysRevLett.126.120604 Introduction.—Symmetryplaysacentralroleinphysics.GivenaquantumsystemdescribedbyHamiltonianoperatorHˆ,asymmetryg,restrictedtobeunitaryinthiswork,isrepresentedbyaunitaryoperatorDˆðgÞ,suchthat ½Hˆ;DˆðgÞ¼0: ð1Þ Ifmultipleg’sformagroupG,Eq.(1)leadstothe fundamentaltheoremthateacheigensubspaceΨE≡fψjHˆψ¼EψgisinvariantunderDˆðgÞforg∈G,oronecancasuallysaythatΨEatleasthassymmetrygroupG.Inotherwords,generally,ΨEhashighersymmetrythanG. Asanexample,considertwo1=2spinscoupledbyaHeisenberginteraction,Hˆ¼Sˆ1·Sˆ
2.ThefullsymmetrygroupofthetripleteigensubspaceisU
(3),ofwhich theHamiltoniansymmetrygroupSO(3)isasubgroup. However,notallsymmetriesinU(3)arephysically interesting,becausemanyoftheminvolvecreating(anni- hilating)entanglementbetweenthespins,andassuchare difficulttorealizeinexperiments.Therefore,hereafterwerestricttomorephysicallyrelevantcases:anoperatorDˆðg˜ÞthatpreservesaneigensubspaceofHˆisconsideredasa“symmetry,”ifandonlyifDˆðg˜Þisadirectproductofunitaryoperatorsonindividualspins;thatis, Dˆðg˜Þ¼dˆ1ðg˜Þ⊗dˆ2ðg˜Þ⊗…⊗dˆNðg˜Þ; ð2Þ knownastheonsite-unitarycondition.ThisrequirestherepresentationofGtobeatensor-productrepresentation,thatis,neitherspatialnortime-reversalsymmetryisconsidered,unlessotherwisespecified.Intheabovetwo-spinexample,aunitaryoperationsendingj↑↑ito pffiffiðj↑↓iþj↓↑iÞ=2leavesthetripleteigensubspaceinvariant,butcannotposeasinEq.
(2).Infact,itcanbe checkedthatallthesymmetriesofthetripleteigensubspace meetingtheonsite-unitaryconditionEq.(2)arejustthe overallrotationsSO
(3).ThetripleteigensubspacehashencethesamesymmetrygroupasHˆitself. Theabovediscussionmotivatesustodefineanewtype ofsymmetryoperation,whichwetentativelytermquasisymmetry,asaunitaryoperatorDˆðg˜ÞsatisfyingEq.
(2),sothatagiveneigensubspaceofHˆhavingenergyEisinvariantunderDˆðg˜Þ.Itisobviousthatg˜’sassuchformanewgroup,denotedbyG˜
E.WecallG˜EthequasisymmetrygroupofHˆwithrespecttotheeigensubspaceΨ
E.IfDˆðg˜ÞmuteswithHˆ,theng˜isaquasisymmetryforanyeigensubspaceofHˆ,sothesymmetrygroupis alwaysasubgroupofanyquasisymmetrygroupforagivenHamiltonian:G⊂G˜
E. Beforeshowinganexplicitexampleofquasisymmetryin quantummodels,wepointoutthatitsclassicalcounterpart, knownasnon-symmetry-causeddegeneracy,iswellknown inmodelsforfrustratedism.ConsideraclassicalJ1−J2modelonasquarelattice,whereHeisenbergJ1couplingsconnectnearestspins,andJ2next-nearestneighborspinsoflengths.ThisHamiltonianisinvariant underanyoverallSO(3)rotation,butisnotinvariantunder relativerotationsbetweenthetwosublattices.Nevertheless, considerastatewhereallspinsineachsublatticeare icallyaligned,thenitiseasytocheckthattheenergy,being−2J2s2perspin,isindependentoftherelativeanglebetweenthetwosublattices.Therefore,a relativerotationbetweenthesublattices,notbeinga 0031-9007=21=126(12)=120604
(6) 120604-
1 ©2021AmericanPhysicalSociety PHYSICALREVIEWLETTERS126,120604(2021) symmetryofH,doesleadtoclassicaldegeneracy.CanweobtainaquasisymmetrymodelbyquantizingtheaboveJ1−J2-model?
Theanswerisnegative:whenquantumfluctuationisturnedon,theaboveclassicaldegeneracyisliftedduetothefamousorder-by-disordermechanism[1]. Wedonotknowadeterministicwayfordiagnosingallpossiblequasi-symmetriesinagivenHamiltonian,quantumorclassical.Yetfortunately,recentprogressinthestudyonquantum-many-bodyscars[2–9]provideswithmanyexamplesofquasisymmetryinquantummodels[10].Incertainnonintegrablequantummany-bodysystems,thereexistsomeclosetrajectoriesintheHilbertspace,alongwhichaspecialshort-range-entangledstateevolvesperiodicallyorquasiperiodically,independentofthesizeofthesystem[12–18].Theevolutionofcertainmany-bodystatesalongtheseclosedtrajectories,asopposedtothechaotictrajectoriesforgenericstates,iscalledthequantummany-bodyscardynamics,orsimplyscardynamics.AllthestatesalongonesuchtrajectoryspanaHilbertsubspaceinvariantundertheHamiltonianevolution,andtheeigenstatesofHˆwithinthissubspaceformatowerofstates,namely,thescartower[19–21].Thescardynamicsisrelatedtotheviolationoftheeigenstate-thermalizationhypothesis[22–26]incertaineigenstatesfromthescartower.Inpreviouslystudiedexactcases[27–33],ascarHamiltonianconsistsoftwoparts, Hˆscar¼HˆþHˆ1; ð3Þ whereHˆhasadegenerateeigensubspaceΨEandHˆ1(i)preservesthesubspaceΨEbut(ii)liftsthedegeneracybybreakingenergyspectrumintoa“tower”withequalspacingδ
E.ItthenesobviousthatarandominitialstateinΨEoscillateswithaperiod2πδE−
1.IfascarHamiltonianinEq.(3)satisfies(i)Hˆ1isasumoflocaloperatorsand(ii)thereisatleastoneproductstateψ0∈ΨE,thenthequantumHamiltonianHˆhasatleastG˜¼Uð1ÞquasisymmetryDˆ½g˜ðθÞ≡expðiHˆ1θÞwithrespecttoΨ
E.Inotherwords,undertheaboveconditions,quantum-many-body-scardynamicsisasufficientcondi- tionfortheexistenceofquasisymmetry. Doesquasisymmetryalsoimplyscardynamics?
SupposethereisaquasisymmetrygroupG˜E≠GforsomeHˆwithrespecttoΨ
E.IfG˜EispactLiegroup,thenthankstotheonsite-unitaryconditionEq.
(2),wehavethat anygenerator Xˆ¼xˆ1⊕xˆ2⊕…⊕xˆ
N ð4Þ isasumoflocaloperatorsxˆi’s,eachofwhichisaHermitianoperatoractingontheithspin.ChooseHˆ1¼cXˆforthescarHamiltonianinEq.
(3),wherecisarealconstant.Foranystateψðt¼0Þ∈ΨEasinitialstate,wehave HˆψðtÞ¼Hˆexp½−iðHˆþHˆ1Þtψðt¼0Þ¼EψðtÞ;ð5Þ meaningthatΨEispreservedbythescarHamiltonianHˆscar.Further,ifXgeneratesaU(1)subgroupofG˜,thenthespectrumofXˆhasequalspacingΔ,andtheevolutionofanyψ∈ΨEhasexactperiod2πðcΔÞ−
1.Therefore,quasisymmetryLiegroupinHˆindeedimpliesscardynamics,giventhatG˜E≠
G.WhenG˜Eisadiscretegroup,thereisnotanobviouschoiceforascarHamiltonian.Inthatcase, thereisadiscreteversionofscardynamics,tobediscussed intheSupplementalMaterial[34].Inthiswork,wefocusonconstructingspinHamiltonians HˆthathaveaquasisymmetrygroupG˜ofchoice.Inthe maintext,weassumethatthequasisymmetrygroupisa compactLiegroup.Ourconstructionschemeusesthree elementsasinput:aspin-sspinchaindefiningtheHilbertspace,s¼1=2;1;3=2;…,pactLiegroupG˜ofchoice,andan“anchorstate,”denotedbyψ0,whichiseitheraproductoramatrix-productstate[37].For simplicity,weinthisworkonlyusetwoanchorstatesas examples:anall-upicstateandanAffleck- Kennedy-Lieb-Tasaki-like[38]matrix-productstate.TheconstructedHamiltonianHˆisexpressedintermsofpro- jectorsactingonsmallclusters,thesameasinRef.[39],but themethodfordefiningthesmall-clusterprojectorsare basedontwoinputs:theanchorstateandthequasisym- metrygroup[40].Productstatesasanchorstates.—Wefirstdescribethe constructionofspin-sHamiltonianswithachosenG˜usingtheall-upstateψ0¼js…siastheanchorstate.Tostartwith,weconsideraclusterofmspins,orsimply,anmcluster.Theproductstateψ0restrictedtoanmclusterisdenotedbyψ½0m.TheunitaryoperatorsonasinglespinformtheunitarygroupUð2sþ1Þ,andweassumethatG˜⊂Uð2sþ1Þ.DefineΨ½G˜masthefollowingsubspaceinthem-clusterspace Ψ½G˜m≡spanfdˆ⊗mðg˜Þψ½0mjg˜∈G˜g; ð6Þ anddefinePˆastheprojectorontoΨ½G˜m.Thenweconsiderthefollowingm-clusterHamiltonian Hˆ½m¼ð1−PˆÞhˆð1−PˆÞ; ð7Þ wherehˆisanarbitraryHermitianmatrixactingonthemcluster.ItiseasytoseethatΨ½G˜misthezeroenergysubspaceofHˆ½mforarandomlychosenhˆ. Now,consideraninfinitechain.ForeachmclusterofconsecutivespinswedefineatermasinEq.
(7),andobtainthefullHamiltonian
X Hˆ¼ ð1−Pˆ½j;jþm−1Þhˆ½j;jþm−1ð1−Pˆ½j;jþm−1Þ;ð8Þ j¼1;…;
N 120604-
2 PHYSICALREVIEWLETTERS126,120604(2021) wherePˆ½j;jþm−1isthem-clusterprojectorinEq.(7)overthej;jþ1;…;jþm−1spins,andhˆ½j;jþm−1isarandom Hermitianoperatoronthesamecluster.ThesummationinEq.(8)isfromj¼1toj¼N−mþ1ifthechainisopen,andtoj¼Nifclosed.Periodiccyclingisunderstoodforaclosedchain:whenjþl>N,replacejþlwithjþl−
N.Twoobservationscanbemade:(i)theall-upstateψ0isazero-energyeigenstateofHˆ,becauseð1−Pˆ½j;jþm−1Þψ0¼0foreachj,and(ii)statesofthe followingform Dˆðg˜Þψ0≡dˆ⊗Nðg˜Þψ
0 ð9Þ arealsozero-energyeigenstatesofHˆforthesamereason.AllDˆðg˜Þψ0’sinEq.(9)andtheirbinationsformasubspaceΨG˜≡spanfDˆðg˜Þψ0jg˜∈G˜g.ItisclearthatΨG˜⊂Ψ0,thezeroenergysubspaceofHˆ.TheHamiltonianHˆhencehasquasisymmetrygroupG˜withrespecttoΨG˜. Tobetterillustratethescheme,welookatoneexamplewheres¼1,m¼2,andG˜¼SOð3Þ⊂Uð3Þ.Forthe 2-cluster,namely,thejthspinandthe(jþ1)thspin,the totalspinS¼0,1,2,andtheall-upstateψ½02¼jþþibelongstoS¼2-subspace.Thereforeactingdˆðg˜Þ⊗dˆðg˜Þ,whereg˜∈SOð3Þonψ½02yieldstheentireS¼2-subspace,whichisΨ½G˜2.The2-clusterprojectorontoΨ½G˜2is Pˆ½j;jþ1¼ðSˆjþSˆjþ1Þ2½ðSˆjþSˆjþ1Þ2−2=24:ð10Þ SubstitutingPˆ½j;jþ1andarandomchoiceforhˆ½j;jþ1intoEq.
(8),wehavethefullHamiltonian.Anexactdiagonal- izationofthisHamiltonian(withperiodicboundary)iscarriedoutfor2≤N≤10.WeplotthelevelstatisticsinRef.[34],whichfitstheWigner-Dysoncurve,indicating nonintegrabilityoftheHamiltonian[42].Thediagonalizationalsoshowsthatthereareexactly2Nþ1independentstatesinΨ0,whicharenothingbutthestatesinthelargesttotalspinsector(totalspinbeingN),andthatΨG˜¼Ψ
0. WecanalsochooseG˜¼SUð2Þ⊂Uð3Þ,andthesameψ0astheanchorstate.InRef.[34],weshowthattheresultantΨG˜(whichagainequalsΨ0)isexactlyspannedby,uptoanonsite-unitarytransform,thetype-Iscar towerofthespin-1-XYmodelinRef.[27],althoughtheHamiltonian,duetotherandomnessinhˆ½j;jþ1,canbedrasticallydifferentfromthatoftheXYmodel.(Thereare twoscartowersdiscoveredinRef.[27],andwedenote them,aftertheirsequentialappearancesintheoriginal paper,astype-Iandtype-II.AlsoseeRef.[29]formoreon thetype-IIcase.) ThissimpleexampleoftheSO(3)quasisymmetrygroup illustratessomegeneralfeaturesofquasisymmetrygroups.First,G˜isasubgroupofUð2sþ1Þ,sothatbychoosingalargesonecanspecifypactLiegroup,suchas SO(n),U(n),Sp(n),andexceptionalLiegroups,asthe quasisymmetrygroup.Wenoteherethattheactualformofthe“sandwiched”partoftheHamiltonianinEq.
(8),hˆi,ispletelyirrelevant,aslongasitdoesnot havesomanysymmetriesthattheHamiltonianes integrable.Last,wewanttoemphasizethat,despitetherandomnessinhˆ½j;jþm−1,itisnotguaranteedthatΨG˜¼Ψ
0.Thisindicatesthatthezero-energysubspaceofHˆ,despitebeingdesignedtobeso,isnotgeneratedbyactingDˆðG˜Þonψ
0.Thisequalitybetweenthetwocanonlybeestablished,ordisproved,innumericsuptosomeN,aswedo inRef.[34].Matrix-productstatesasanchorstates.—Aproductstate haszeroentanglement,andifchosenastheanchorstate, or,equivalently,theinitialstate,duringthetimeevolution thestateremainsaproductstate,becausequasisymmetry operationsarestrictlylocal.Itisnaturalthatweextend thediscussiontothecasewheretheanchorstatehas finiteentanglement;i.e.,isamatrix-productstate.The correspondingconstructionofthescarHamiltonian followsaslightlyplicatedscheme,pared withtheproduct-statecase.AgainconsideringagroupG˜⊂Uð2sþ1Þ,wefirstobtaintwolinearorprojectiverepresentationsofVofequaldimensionχ,dLðG˜Þ,dRðG˜Þ,suchthatdL⊗dRcontainsarepresentationofdimension2sþ1,denotedbydðG˜Þ.Inotherwords,thereexistsatrioofrepresentationsdL,dR,dofdimensionsχ,χ,and2sþ1,suchthattheClebsch-GordoncoefficientshdL;α;dR;βjd;ki≠0,whereα;β¼1;…;χandk¼1;…;2sþ
1.Whentheseconditionsaremet,definematrices(Fig.1showshowquasisymmetriesactonthese matrices) Akαβ≡hdL;α;dR;βjd;ki: ð11Þ Thesematricesdefineouranchorstate(s),whichis ψ0¼TrðAs1…AsNÞjs1;…;sNi; ψαβ¼ðAs1…AsNÞαβjs1;…;sNi; ð12Þ foraclosedandanopenchain,respectively. Consideranmcluster,onwhichthematricesEq.(11)defineχ2open-matrix-productstates FIG.1.Actionofonsiteoperatordˆðg˜ÞontheClebsch-GordoncoefficientstensorA;therepresentationdðG˜Þonthephysicalindicesistransferredtotwo(projective)representationsdL;RðG˜Þonthebondindices. 120604-
3 PHYSICALREVIEWLETTERS126,120604(2021) ψ½αmβ¼ðAs1…AsmÞαβjs1;…;smi; ð13Þ whereα;β¼1;…;χ.Actingdˆ⊗mðg˜Þforanyg˜∈G˜ontheseχ2statesyieldsanothersetofχ2open-matrix-product states: hs1…smjdˆ⊗mðg˜Þψ½αmβi ≡dss0ðg˜Þ…dss0ðg˜Þ½As01…As0mαβ 11 mm ¼½dLðg˜ÞAs1dTRðg˜Þ…dLðg˜ÞAsmdTRðg˜Þαβ: ð14Þ Findthesubspace Ψ½G˜m≡spanfdˆ⊗mðg˜Þψ½αmβjg˜∈G˜;α;β¼1;…;χg;ð15Þ anddefinePˆastheprojectorontoΨ½G˜m.ForaclosedchainofN≥msites,definethe HamiltonianasinEq.
(8).Itiseasytoverifythattheanchorstateψ0isazeroeigenstateofHˆbecauseitisazeroeigenstateofeachterm;andalsothestateDˆðg˜Þψ0≡dˆ⊗Nðg˜Þψ0isazeroeigenstateforthesamereasonforg˜∈G˜.ThespaceΨG˜spannedbyallthesestatesisthusazero-energysubspaceofHˆ,i.e.,ΨG˜⊂Ψ
0.Therefore,wehaveconstructedHˆthathasquasisymmetrygroupG˜withrespecttoΨG˜.Thecaseofopenchainscanbesimilarlyworkedout(notshownhere). Weagainuseanexampletoillustratetheabovecon- structionscheme.ChooseG˜¼Uð1Þ⊂Uð3Þasourqua- sigroup, and we choose dLðG˜Þ ¼ dRðG˜Þ ¼ 12 ⊕ − 12, which arethetwo-dimensionalreducibleprojectiverepresenta- tionsofU
(1).ThespecificrealizationofU(1)canbe arbitrary,butinthisexamplewechooseittobetheoverallspinrotationaboutthezaxis.dðG˜Þischosentobe thethree-dimensionalreduciblevectorrepresentationdðG˜Þ¼ðx;y;zÞ¼þ1⊕0⊕−
1.Sothematricesare givenbytheClebsch-Gordoncoefficients rffiffi rffiffi AƼ1ðσ0ÆσzÞ;A0¼1σx;ð16Þ
6 3 satisfying expðiSˆzθÞijAj¼eiσzθ=2Aiðeiσzθ=2ÞT: ð17Þ Nowweconsideranm¼3-cluster.Thefouropen3-clusterstates,ψαβ,arenonebuttheAffleck-Kennedy-Lieb-Tasakiopen3-chaingroundstates,uptoaunitarytransformexpðiSyπÞonalloddsites. AfteractingallelementsoftheU(1)quasisymmetry grouponthefouropen3-clusterstates,wehaveasubspaceΨ½G˜3spannedby12states,classifiedintogroupslabeledbytwoquantumnumbersnÆ≡Sˆ1zÆSˆ2zþS3z: ðnþ;n−Þ¼ð0;0Þ∶jþ0−ip−ffiffij−0þi;jþ0−iþjp−ffi0ffiþiþj000i;
2 3 ðÆ1;Æ1Þ∶jÆ00ipþffiffij00Æi;
2 ðÆ1;∓1Þ∶j0Æ0i; ðÆ2;0Þ∶jÆÆ0i;j0ÆÆi; ðÆ3;Æ1Þ∶jÆÆÆi: ð18Þ DefinePˆasthe3-clusterprojectorontoΨ½G˜3.ReplacingPˆ½j;jþ2withPˆinEq.
(8),wehavethefullHamiltonianHˆwithquasisymmetryU
(1),withrespecttothezeroenergysubspaceΨG˜.UsingnumericalcalculationuptoN¼10sites[34],wefindthatlevel-spacingstatisticsofHˆshowsWignerDysonbehavior.Wehavealsochecked,uptoN¼14,thatthedegeneracyofthezerosubspaceofHˆisNþ1forperiodicchainsand4Nforopenchains,andthatΨ0¼ΨG˜.Thismeansthattheentirezero-energysubspaceofHˆcanbeobtainedfromactingthequasisymmetrygroupelementsontheanchorstate(s).Itisinterestingtonoticethat,afteranonsite-unitarytransform,theresultantzero-energysubspaceesthespacespannedbythetype-II-spin-1-XYscar[27,29].mentthatsincethequasisymmetrygroupisonlyU
(1),insteadofSU(2)or higherLiegroups,thereisnotanobviouschoiceforalocalQˆsuchthat½Qˆ;Hˆ¼constÃQˆonthesubspace.WealsoremarkthattheHamiltonianfollowingourconstructionis“unfrustrated,”inthesensethatΨ0lieswithinthezeroenergysubspaceofeachterminHˆ,incontrasttotheoriginalXYmodel.Itiscertainlypossibletoconstructmodelshavinglargerquasisymmetrygroups,suchasSO
(3),usingthesameMPSasinEq.(16),anexplicitexampleofwhichisshowninRef.[34]. mentthatusingmatrix-productstatesasanchor statesisparticularlyusefulwhenwerelatethisstudytothestudyofologicalstates[43–45] (SPT).IntheSupplementalMaterial[34],weshowhow onecanconstructascartowerandHamiltoniansuchthatallstatesoftheformDˆðg˜Þψ0isanSPTprotectedbyaunitary 120604-
4 PHYSICALREVIEWLETTERS126,120604(2021) orantiunitarygroup.Herewesimplypointoutthatintheexampleabove,bothψ0andDˆðg˜Þψ0areSPTprotectedbytime-reversalsymmetry,demonstratedinRef.[34]. Discussion.—Aimingforasimplenarrative,wehavesofarassumedthattheanchorstateshavetranslationsymmetry,andthequasi-groupsymmetryoperatorDˆðg˜Þactsuniformlyoneachspin,asinEq.
(9).Bothconditionscanberelaxed:(i)theanchorstatemayberotatedbyonsite-unitaryoperatorsdˆ1ðg˜1Þ⊗dˆ2ðg˜2Þ⊗…⊗dˆNðg˜NÞforg˜i∈G˜;(ii)theactionofDˆðG˜Þcanbegeneralizedto Dˆðg˜Þ¼dˆ1ðg˜Þ⊗dˆ2ðg˜Þ⊗…⊗dˆNðg˜Þ;ð19Þ wheredˆi¼1;…;NareNdifferentrepresentationsofG˜.Withthesegeneralizations,themethodfordefiningthem-clusterprojectorsesslightlymodified,showninRef.[34]. Theanchorstate,product,ormatrixproduct,isakeyinputforourconstructionscheme.Itensuresthatwithinthezero-energysubspaceofconstructedHamiltonian,thereisatleastonestatethatisa(matrix)productstate.Theanchorstatecanalsobeusedastheinitialstateintheassociatedscardynamics,andduetotheonsite-unitarycondition,allthestatesalongtheentiretrajectoryare(matrix)productstatesastheanchorstate.Inpreviousstudies,thestateusedastheorigin,fromwhichthescartowerisobtainedusingladderoperators,isanexacteigenstateofthescarHamiltonian,ratherthananinitialstateforscardynamics. WeimposethequasisymmetrygroupG˜withoutrequiringaladderoperatorQˆ.However,ifG˜isanon-AbelianLiegroup,aladderoperatorcanalwaysbefound,becauseinthatcaseSOð3Þ⊂G˜,andSO(3)hasladderoperatorQˆ¼Lˆx−iLˆy.ForG˜¼Uð1Þ,wehaveusedoneaboveexampletoshowthatevenintheabsenceofQ,thezeroenergysubspaceofHˆformsascartoweridenticaltothetype-II-spin-1-XYscartower.Ontheotherhand,ifG˜⊃SOð3Þ,thereare,ingeneral,multipleladderoperators.Forexample,whenG˜¼SUð3Þ⊃SUð2Þ,therearethreedifferentladderoperators,correspondingtothethreenaturalembeddingsofSU(2)inSU
(3).SeeRef.[34]foranexplicitmodel,andageneraldiscussionontherelationbetweenthenon-Abelianquasisymmetrygroupandladderoperators. Tosummarize,weshowthatmany-body-scartowershavehiddengroupstructuresthatwecallquasisymmetrygroups,andproposeschemesforconstructinglocalHamiltoniansthathostanychosenLiegroupasitsquasisymmetrygroup.Asapplicationofthenewconcept,weshowthat(i)severalknownscarmodelscanbeunified,(ii)ascarmodelhavingthreesetsofladderoperatorscanbefound(seeRef.[34]),and(iii)adiscreteversionofmanybodyscarisestablishedbychoosingadiscretequasisymmetrygroup(seeRef.[34]).
C.F.thanksB.A.Bernevigfordiscussionandfeedback,andthanksHuanHeforthefirstintroductiontothestudyof quantummany-bodyscars.C.L.andJ.R.thankShuChenforhisencouragementandsupportattheearlystageofthisproject.C.F.andJ.R.acknowledgesupportfromMinistryofScienceandTechnologyofChinaunderGrantNo.2016YFA0302400,NationalScienceFoundationofChinaunderGrantNo.11674370,andChineseAcademyofSciencesunderGrantNo.XXH13506-202andXDB33000000.C.L.acknowledgessupportfromMinistryofScienceandTechnologyofChinaunderGrantNo.2016YFA0300600. *Theseauthorscontributedequallytothiswork.†cfang@[1]
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6
2,*ChenguangLiang,1,
2,*andChenFang1,3,
4,† 1BeijingNationalLaboratoryforCondensedMatterPhysicsandInstituteofPhysics,ChineseAcademyofSciences,Beijing100190,China 2UniversityofChineseAcademyofSciences,Beijing100049,China3SongshanLakeMaterialsLaboratory,Dongguan,Guangdong523808,China4KavliInstituteforTheoreticalSciences,ChineseAcademyofSciences,Beijing100190,China (Received29July2020;revised6December2020;epted24February2021;published24March2021) Inquantumsystems,asubspacespannedbydegenerateeigenvectorsoftheHamiltonianmayhavehighersymmetriesthanthoseoftheHamiltonianitself.Whenthisenhanced-symmetrygroupcanbegeneratedfromlocaloperators,wecallitaquasisymmetrygroup.WhenthegroupisaLiegroup,anexternalfieldcoupledtocertaingeneratorsofthequasisymmetrygroupliftsthedegeneracy,andresultsinexactlyperiodicdynamicswithinthedegeneratesubspace,namely,themany-body-scardynamics(giventhatHamiltonianisnonintegrable).Weprovidetworelatedschemesforconstructingone-dimensionalspinmodelshavingon-demandquasisymmetrygroups,withexactperiodicevolutionofaprechosenproductormatrix-productstateunderexternalfields. DOI:10.1103/PhysRevLett.126.120604 Introduction.—Symmetryplaysacentralroleinphysics.GivenaquantumsystemdescribedbyHamiltonianoperatorHˆ,asymmetryg,restrictedtobeunitaryinthiswork,isrepresentedbyaunitaryoperatorDˆðgÞ,suchthat ½Hˆ;DˆðgÞ¼0: ð1Þ Ifmultipleg’sformagroupG,Eq.(1)leadstothe fundamentaltheoremthateacheigensubspaceΨE≡fψjHˆψ¼EψgisinvariantunderDˆðgÞforg∈G,oronecancasuallysaythatΨEatleasthassymmetrygroupG.Inotherwords,generally,ΨEhashighersymmetrythanG. Asanexample,considertwo1=2spinscoupledbyaHeisenberginteraction,Hˆ¼Sˆ1·Sˆ
2.ThefullsymmetrygroupofthetripleteigensubspaceisU
(3),ofwhich theHamiltoniansymmetrygroupSO(3)isasubgroup. However,notallsymmetriesinU(3)arephysically interesting,becausemanyoftheminvolvecreating(anni- hilating)entanglementbetweenthespins,andassuchare difficulttorealizeinexperiments.Therefore,hereafterwerestricttomorephysicallyrelevantcases:anoperatorDˆðg˜ÞthatpreservesaneigensubspaceofHˆisconsideredasa“symmetry,”ifandonlyifDˆðg˜Þisadirectproductofunitaryoperatorsonindividualspins;thatis, Dˆðg˜Þ¼dˆ1ðg˜Þ⊗dˆ2ðg˜Þ⊗…⊗dˆNðg˜Þ; ð2Þ knownastheonsite-unitarycondition.ThisrequirestherepresentationofGtobeatensor-productrepresentation,thatis,neitherspatialnortime-reversalsymmetryisconsidered,unlessotherwisespecified.Intheabovetwo-spinexample,aunitaryoperationsendingj↑↑ito pffiffiðj↑↓iþj↓↑iÞ=2leavesthetripleteigensubspaceinvariant,butcannotposeasinEq.
(2).Infact,itcanbe checkedthatallthesymmetriesofthetripleteigensubspace meetingtheonsite-unitaryconditionEq.(2)arejustthe overallrotationsSO
(3).ThetripleteigensubspacehashencethesamesymmetrygroupasHˆitself. Theabovediscussionmotivatesustodefineanewtype ofsymmetryoperation,whichwetentativelytermquasisymmetry,asaunitaryoperatorDˆðg˜ÞsatisfyingEq.
(2),sothatagiveneigensubspaceofHˆhavingenergyEisinvariantunderDˆðg˜Þ.Itisobviousthatg˜’sassuchformanewgroup,denotedbyG˜
E.WecallG˜EthequasisymmetrygroupofHˆwithrespecttotheeigensubspaceΨ
E.IfDˆðg˜ÞmuteswithHˆ,theng˜isaquasisymmetryforanyeigensubspaceofHˆ,sothesymmetrygroupis alwaysasubgroupofanyquasisymmetrygroupforagivenHamiltonian:G⊂G˜
E. Beforeshowinganexplicitexampleofquasisymmetryin quantummodels,wepointoutthatitsclassicalcounterpart, knownasnon-symmetry-causeddegeneracy,iswellknown inmodelsforfrustratedism.ConsideraclassicalJ1−J2modelonasquarelattice,whereHeisenbergJ1couplingsconnectnearestspins,andJ2next-nearestneighborspinsoflengths.ThisHamiltonianisinvariant underanyoverallSO(3)rotation,butisnotinvariantunder relativerotationsbetweenthetwosublattices.Nevertheless, considerastatewhereallspinsineachsublatticeare icallyaligned,thenitiseasytocheckthattheenergy,being−2J2s2perspin,isindependentoftherelativeanglebetweenthetwosublattices.Therefore,a relativerotationbetweenthesublattices,notbeinga 0031-9007=21=126(12)=120604
(6) 120604-
1 ©2021AmericanPhysicalSociety PHYSICALREVIEWLETTERS126,120604(2021) symmetryofH,doesleadtoclassicaldegeneracy.CanweobtainaquasisymmetrymodelbyquantizingtheaboveJ1−J2-model?
Theanswerisnegative:whenquantumfluctuationisturnedon,theaboveclassicaldegeneracyisliftedduetothefamousorder-by-disordermechanism[1]. Wedonotknowadeterministicwayfordiagnosingallpossiblequasi-symmetriesinagivenHamiltonian,quantumorclassical.Yetfortunately,recentprogressinthestudyonquantum-many-bodyscars[2–9]provideswithmanyexamplesofquasisymmetryinquantummodels[10].Incertainnonintegrablequantummany-bodysystems,thereexistsomeclosetrajectoriesintheHilbertspace,alongwhichaspecialshort-range-entangledstateevolvesperiodicallyorquasiperiodically,independentofthesizeofthesystem[12–18].Theevolutionofcertainmany-bodystatesalongtheseclosedtrajectories,asopposedtothechaotictrajectoriesforgenericstates,iscalledthequantummany-bodyscardynamics,orsimplyscardynamics.AllthestatesalongonesuchtrajectoryspanaHilbertsubspaceinvariantundertheHamiltonianevolution,andtheeigenstatesofHˆwithinthissubspaceformatowerofstates,namely,thescartower[19–21].Thescardynamicsisrelatedtotheviolationoftheeigenstate-thermalizationhypothesis[22–26]incertaineigenstatesfromthescartower.Inpreviouslystudiedexactcases[27–33],ascarHamiltonianconsistsoftwoparts, Hˆscar¼HˆþHˆ1; ð3Þ whereHˆhasadegenerateeigensubspaceΨEandHˆ1(i)preservesthesubspaceΨEbut(ii)liftsthedegeneracybybreakingenergyspectrumintoa“tower”withequalspacingδ
E.ItthenesobviousthatarandominitialstateinΨEoscillateswithaperiod2πδE−
1.IfascarHamiltonianinEq.(3)satisfies(i)Hˆ1isasumoflocaloperatorsand(ii)thereisatleastoneproductstateψ0∈ΨE,thenthequantumHamiltonianHˆhasatleastG˜¼Uð1ÞquasisymmetryDˆ½g˜ðθÞ≡expðiHˆ1θÞwithrespecttoΨ
E.Inotherwords,undertheaboveconditions,quantum-many-body-scardynamicsisasufficientcondi- tionfortheexistenceofquasisymmetry. Doesquasisymmetryalsoimplyscardynamics?
SupposethereisaquasisymmetrygroupG˜E≠GforsomeHˆwithrespecttoΨ
E.IfG˜EispactLiegroup,thenthankstotheonsite-unitaryconditionEq.
(2),wehavethat anygenerator Xˆ¼xˆ1⊕xˆ2⊕…⊕xˆ
N ð4Þ isasumoflocaloperatorsxˆi’s,eachofwhichisaHermitianoperatoractingontheithspin.ChooseHˆ1¼cXˆforthescarHamiltonianinEq.
(3),wherecisarealconstant.Foranystateψðt¼0Þ∈ΨEasinitialstate,wehave HˆψðtÞ¼Hˆexp½−iðHˆþHˆ1Þtψðt¼0Þ¼EψðtÞ;ð5Þ meaningthatΨEispreservedbythescarHamiltonianHˆscar.Further,ifXgeneratesaU(1)subgroupofG˜,thenthespectrumofXˆhasequalspacingΔ,andtheevolutionofanyψ∈ΨEhasexactperiod2πðcΔÞ−
1.Therefore,quasisymmetryLiegroupinHˆindeedimpliesscardynamics,giventhatG˜E≠
G.WhenG˜Eisadiscretegroup,thereisnotanobviouschoiceforascarHamiltonian.Inthatcase, thereisadiscreteversionofscardynamics,tobediscussed intheSupplementalMaterial[34].Inthiswork,wefocusonconstructingspinHamiltonians HˆthathaveaquasisymmetrygroupG˜ofchoice.Inthe maintext,weassumethatthequasisymmetrygroupisa compactLiegroup.Ourconstructionschemeusesthree elementsasinput:aspin-sspinchaindefiningtheHilbertspace,s¼1=2;1;3=2;…,pactLiegroupG˜ofchoice,andan“anchorstate,”denotedbyψ0,whichiseitheraproductoramatrix-productstate[37].For simplicity,weinthisworkonlyusetwoanchorstatesas examples:anall-upicstateandanAffleck- Kennedy-Lieb-Tasaki-like[38]matrix-productstate.TheconstructedHamiltonianHˆisexpressedintermsofpro- jectorsactingonsmallclusters,thesameasinRef.[39],but themethodfordefiningthesmall-clusterprojectorsare basedontwoinputs:theanchorstateandthequasisym- metrygroup[40].Productstatesasanchorstates.—Wefirstdescribethe constructionofspin-sHamiltonianswithachosenG˜usingtheall-upstateψ0¼js…siastheanchorstate.Tostartwith,weconsideraclusterofmspins,orsimply,anmcluster.Theproductstateψ0restrictedtoanmclusterisdenotedbyψ½0m.TheunitaryoperatorsonasinglespinformtheunitarygroupUð2sþ1Þ,andweassumethatG˜⊂Uð2sþ1Þ.DefineΨ½G˜masthefollowingsubspaceinthem-clusterspace Ψ½G˜m≡spanfdˆ⊗mðg˜Þψ½0mjg˜∈G˜g; ð6Þ anddefinePˆastheprojectorontoΨ½G˜m.Thenweconsiderthefollowingm-clusterHamiltonian Hˆ½m¼ð1−PˆÞhˆð1−PˆÞ; ð7Þ wherehˆisanarbitraryHermitianmatrixactingonthemcluster.ItiseasytoseethatΨ½G˜misthezeroenergysubspaceofHˆ½mforarandomlychosenhˆ. Now,consideraninfinitechain.ForeachmclusterofconsecutivespinswedefineatermasinEq.
(7),andobtainthefullHamiltonian
X Hˆ¼ ð1−Pˆ½j;jþm−1Þhˆ½j;jþm−1ð1−Pˆ½j;jþm−1Þ;ð8Þ j¼1;…;
N 120604-
2 PHYSICALREVIEWLETTERS126,120604(2021) wherePˆ½j;jþm−1isthem-clusterprojectorinEq.(7)overthej;jþ1;…;jþm−1spins,andhˆ½j;jþm−1isarandom Hermitianoperatoronthesamecluster.ThesummationinEq.(8)isfromj¼1toj¼N−mþ1ifthechainisopen,andtoj¼Nifclosed.Periodiccyclingisunderstoodforaclosedchain:whenjþl>N,replacejþlwithjþl−
N.Twoobservationscanbemade:(i)theall-upstateψ0isazero-energyeigenstateofHˆ,becauseð1−Pˆ½j;jþm−1Þψ0¼0foreachj,and(ii)statesofthe followingform Dˆðg˜Þψ0≡dˆ⊗Nðg˜Þψ
0 ð9Þ arealsozero-energyeigenstatesofHˆforthesamereason.AllDˆðg˜Þψ0’sinEq.(9)andtheirbinationsformasubspaceΨG˜≡spanfDˆðg˜Þψ0jg˜∈G˜g.ItisclearthatΨG˜⊂Ψ0,thezeroenergysubspaceofHˆ.TheHamiltonianHˆhencehasquasisymmetrygroupG˜withrespecttoΨG˜. Tobetterillustratethescheme,welookatoneexamplewheres¼1,m¼2,andG˜¼SOð3Þ⊂Uð3Þ.Forthe 2-cluster,namely,thejthspinandthe(jþ1)thspin,the totalspinS¼0,1,2,andtheall-upstateψ½02¼jþþibelongstoS¼2-subspace.Thereforeactingdˆðg˜Þ⊗dˆðg˜Þ,whereg˜∈SOð3Þonψ½02yieldstheentireS¼2-subspace,whichisΨ½G˜2.The2-clusterprojectorontoΨ½G˜2is Pˆ½j;jþ1¼ðSˆjþSˆjþ1Þ2½ðSˆjþSˆjþ1Þ2−2=24:ð10Þ SubstitutingPˆ½j;jþ1andarandomchoiceforhˆ½j;jþ1intoEq.
(8),wehavethefullHamiltonian.Anexactdiagonal- izationofthisHamiltonian(withperiodicboundary)iscarriedoutfor2≤N≤10.WeplotthelevelstatisticsinRef.[34],whichfitstheWigner-Dysoncurve,indicating nonintegrabilityoftheHamiltonian[42].Thediagonalizationalsoshowsthatthereareexactly2Nþ1independentstatesinΨ0,whicharenothingbutthestatesinthelargesttotalspinsector(totalspinbeingN),andthatΨG˜¼Ψ
0. WecanalsochooseG˜¼SUð2Þ⊂Uð3Þ,andthesameψ0astheanchorstate.InRef.[34],weshowthattheresultantΨG˜(whichagainequalsΨ0)isexactlyspannedby,uptoanonsite-unitarytransform,thetype-Iscar towerofthespin-1-XYmodelinRef.[27],althoughtheHamiltonian,duetotherandomnessinhˆ½j;jþ1,canbedrasticallydifferentfromthatoftheXYmodel.(Thereare twoscartowersdiscoveredinRef.[27],andwedenote them,aftertheirsequentialappearancesintheoriginal paper,astype-Iandtype-II.AlsoseeRef.[29]formoreon thetype-IIcase.) ThissimpleexampleoftheSO(3)quasisymmetrygroup illustratessomegeneralfeaturesofquasisymmetrygroups.First,G˜isasubgroupofUð2sþ1Þ,sothatbychoosingalargesonecanspecifypactLiegroup,suchas SO(n),U(n),Sp(n),andexceptionalLiegroups,asthe quasisymmetrygroup.Wenoteherethattheactualformofthe“sandwiched”partoftheHamiltonianinEq.
(8),hˆi,ispletelyirrelevant,aslongasitdoesnot havesomanysymmetriesthattheHamiltonianes integrable.Last,wewanttoemphasizethat,despitetherandomnessinhˆ½j;jþm−1,itisnotguaranteedthatΨG˜¼Ψ
0.Thisindicatesthatthezero-energysubspaceofHˆ,despitebeingdesignedtobeso,isnotgeneratedbyactingDˆðG˜Þonψ
0.Thisequalitybetweenthetwocanonlybeestablished,ordisproved,innumericsuptosomeN,aswedo inRef.[34].Matrix-productstatesasanchorstates.—Aproductstate haszeroentanglement,andifchosenastheanchorstate, or,equivalently,theinitialstate,duringthetimeevolution thestateremainsaproductstate,becausequasisymmetry operationsarestrictlylocal.Itisnaturalthatweextend thediscussiontothecasewheretheanchorstatehas finiteentanglement;i.e.,isamatrix-productstate.The correspondingconstructionofthescarHamiltonian followsaslightlyplicatedscheme,pared withtheproduct-statecase.AgainconsideringagroupG˜⊂Uð2sþ1Þ,wefirstobtaintwolinearorprojectiverepresentationsofVofequaldimensionχ,dLðG˜Þ,dRðG˜Þ,suchthatdL⊗dRcontainsarepresentationofdimension2sþ1,denotedbydðG˜Þ.Inotherwords,thereexistsatrioofrepresentationsdL,dR,dofdimensionsχ,χ,and2sþ1,suchthattheClebsch-GordoncoefficientshdL;α;dR;βjd;ki≠0,whereα;β¼1;…;χandk¼1;…;2sþ
1.Whentheseconditionsaremet,definematrices(Fig.1showshowquasisymmetriesactonthese matrices) Akαβ≡hdL;α;dR;βjd;ki: ð11Þ Thesematricesdefineouranchorstate(s),whichis ψ0¼TrðAs1…AsNÞjs1;…;sNi; ψαβ¼ðAs1…AsNÞαβjs1;…;sNi; ð12Þ foraclosedandanopenchain,respectively. Consideranmcluster,onwhichthematricesEq.(11)defineχ2open-matrix-productstates FIG.1.Actionofonsiteoperatordˆðg˜ÞontheClebsch-GordoncoefficientstensorA;therepresentationdðG˜Þonthephysicalindicesistransferredtotwo(projective)representationsdL;RðG˜Þonthebondindices. 120604-
3 PHYSICALREVIEWLETTERS126,120604(2021) ψ½αmβ¼ðAs1…AsmÞαβjs1;…;smi; ð13Þ whereα;β¼1;…;χ.Actingdˆ⊗mðg˜Þforanyg˜∈G˜ontheseχ2statesyieldsanothersetofχ2open-matrix-product states: hs1…smjdˆ⊗mðg˜Þψ½αmβi ≡dss0ðg˜Þ…dss0ðg˜Þ½As01…As0mαβ 11 mm ¼½dLðg˜ÞAs1dTRðg˜Þ…dLðg˜ÞAsmdTRðg˜Þαβ: ð14Þ Findthesubspace Ψ½G˜m≡spanfdˆ⊗mðg˜Þψ½αmβjg˜∈G˜;α;β¼1;…;χg;ð15Þ anddefinePˆastheprojectorontoΨ½G˜m.ForaclosedchainofN≥msites,definethe HamiltonianasinEq.
(8).Itiseasytoverifythattheanchorstateψ0isazeroeigenstateofHˆbecauseitisazeroeigenstateofeachterm;andalsothestateDˆðg˜Þψ0≡dˆ⊗Nðg˜Þψ0isazeroeigenstateforthesamereasonforg˜∈G˜.ThespaceΨG˜spannedbyallthesestatesisthusazero-energysubspaceofHˆ,i.e.,ΨG˜⊂Ψ
0.Therefore,wehaveconstructedHˆthathasquasisymmetrygroupG˜withrespecttoΨG˜.Thecaseofopenchainscanbesimilarlyworkedout(notshownhere). Weagainuseanexampletoillustratetheabovecon- structionscheme.ChooseG˜¼Uð1Þ⊂Uð3Þasourqua- sigroup, and we choose dLðG˜Þ ¼ dRðG˜Þ ¼ 12 ⊕ − 12, which arethetwo-dimensionalreducibleprojectiverepresenta- tionsofU
(1).ThespecificrealizationofU(1)canbe arbitrary,butinthisexamplewechooseittobetheoverallspinrotationaboutthezaxis.dðG˜Þischosentobe thethree-dimensionalreduciblevectorrepresentationdðG˜Þ¼ðx;y;zÞ¼þ1⊕0⊕−
1.Sothematricesare givenbytheClebsch-Gordoncoefficients rffiffi rffiffi AƼ1ðσ0ÆσzÞ;A0¼1σx;ð16Þ
6 3 satisfying expðiSˆzθÞijAj¼eiσzθ=2Aiðeiσzθ=2ÞT: ð17Þ Nowweconsideranm¼3-cluster.Thefouropen3-clusterstates,ψαβ,arenonebuttheAffleck-Kennedy-Lieb-Tasakiopen3-chaingroundstates,uptoaunitarytransformexpðiSyπÞonalloddsites. AfteractingallelementsoftheU(1)quasisymmetry grouponthefouropen3-clusterstates,wehaveasubspaceΨ½G˜3spannedby12states,classifiedintogroupslabeledbytwoquantumnumbersnÆ≡Sˆ1zÆSˆ2zþS3z: ðnþ;n−Þ¼ð0;0Þ∶jþ0−ip−ffiffij−0þi;jþ0−iþjp−ffi0ffiþiþj000i;
2 3 ðÆ1;Æ1Þ∶jÆ00ipþffiffij00Æi;
2 ðÆ1;∓1Þ∶j0Æ0i; ðÆ2;0Þ∶jÆÆ0i;j0ÆÆi; ðÆ3;Æ1Þ∶jÆÆÆi: ð18Þ DefinePˆasthe3-clusterprojectorontoΨ½G˜3.ReplacingPˆ½j;jþ2withPˆinEq.
(8),wehavethefullHamiltonianHˆwithquasisymmetryU
(1),withrespecttothezeroenergysubspaceΨG˜.UsingnumericalcalculationuptoN¼10sites[34],wefindthatlevel-spacingstatisticsofHˆshowsWignerDysonbehavior.Wehavealsochecked,uptoN¼14,thatthedegeneracyofthezerosubspaceofHˆisNþ1forperiodicchainsand4Nforopenchains,andthatΨ0¼ΨG˜.Thismeansthattheentirezero-energysubspaceofHˆcanbeobtainedfromactingthequasisymmetrygroupelementsontheanchorstate(s).Itisinterestingtonoticethat,afteranonsite-unitarytransform,theresultantzero-energysubspaceesthespacespannedbythetype-II-spin-1-XYscar[27,29].mentthatsincethequasisymmetrygroupisonlyU
(1),insteadofSU(2)or higherLiegroups,thereisnotanobviouschoiceforalocalQˆsuchthat½Qˆ;Hˆ¼constÃQˆonthesubspace.WealsoremarkthattheHamiltonianfollowingourconstructionis“unfrustrated,”inthesensethatΨ0lieswithinthezeroenergysubspaceofeachterminHˆ,incontrasttotheoriginalXYmodel.Itiscertainlypossibletoconstructmodelshavinglargerquasisymmetrygroups,suchasSO
(3),usingthesameMPSasinEq.(16),anexplicitexampleofwhichisshowninRef.[34]. mentthatusingmatrix-productstatesasanchor statesisparticularlyusefulwhenwerelatethisstudytothestudyofologicalstates[43–45] (SPT).IntheSupplementalMaterial[34],weshowhow onecanconstructascartowerandHamiltoniansuchthatallstatesoftheformDˆðg˜Þψ0isanSPTprotectedbyaunitary 120604-
4 PHYSICALREVIEWLETTERS126,120604(2021) orantiunitarygroup.Herewesimplypointoutthatintheexampleabove,bothψ0andDˆðg˜Þψ0areSPTprotectedbytime-reversalsymmetry,demonstratedinRef.[34]. Discussion.—Aimingforasimplenarrative,wehavesofarassumedthattheanchorstateshavetranslationsymmetry,andthequasi-groupsymmetryoperatorDˆðg˜Þactsuniformlyoneachspin,asinEq.
(9).Bothconditionscanberelaxed:(i)theanchorstatemayberotatedbyonsite-unitaryoperatorsdˆ1ðg˜1Þ⊗dˆ2ðg˜2Þ⊗…⊗dˆNðg˜NÞforg˜i∈G˜;(ii)theactionofDˆðG˜Þcanbegeneralizedto Dˆðg˜Þ¼dˆ1ðg˜Þ⊗dˆ2ðg˜Þ⊗…⊗dˆNðg˜Þ;ð19Þ wheredˆi¼1;…;NareNdifferentrepresentationsofG˜.Withthesegeneralizations,themethodfordefiningthem-clusterprojectorsesslightlymodified,showninRef.[34]. Theanchorstate,product,ormatrixproduct,isakeyinputforourconstructionscheme.Itensuresthatwithinthezero-energysubspaceofconstructedHamiltonian,thereisatleastonestatethatisa(matrix)productstate.Theanchorstatecanalsobeusedastheinitialstateintheassociatedscardynamics,andduetotheonsite-unitarycondition,allthestatesalongtheentiretrajectoryare(matrix)productstatesastheanchorstate.Inpreviousstudies,thestateusedastheorigin,fromwhichthescartowerisobtainedusingladderoperators,isanexacteigenstateofthescarHamiltonian,ratherthananinitialstateforscardynamics. WeimposethequasisymmetrygroupG˜withoutrequiringaladderoperatorQˆ.However,ifG˜isanon-AbelianLiegroup,aladderoperatorcanalwaysbefound,becauseinthatcaseSOð3Þ⊂G˜,andSO(3)hasladderoperatorQˆ¼Lˆx−iLˆy.ForG˜¼Uð1Þ,wehaveusedoneaboveexampletoshowthatevenintheabsenceofQ,thezeroenergysubspaceofHˆformsascartoweridenticaltothetype-II-spin-1-XYscartower.Ontheotherhand,ifG˜⊃SOð3Þ,thereare,ingeneral,multipleladderoperators.Forexample,whenG˜¼SUð3Þ⊃SUð2Þ,therearethreedifferentladderoperators,correspondingtothethreenaturalembeddingsofSU(2)inSU
(3).SeeRef.[34]foranexplicitmodel,andageneraldiscussionontherelationbetweenthenon-Abelianquasisymmetrygroupandladderoperators. Tosummarize,weshowthatmany-body-scartowershavehiddengroupstructuresthatwecallquasisymmetrygroups,andproposeschemesforconstructinglocalHamiltoniansthathostanychosenLiegroupasitsquasisymmetrygroup.Asapplicationofthenewconcept,weshowthat(i)severalknownscarmodelscanbeunified,(ii)ascarmodelhavingthreesetsofladderoperatorscanbefound(seeRef.[34]),and(iii)adiscreteversionofmanybodyscarisestablishedbychoosingadiscretequasisymmetrygroup(seeRef.[34]).
C.F.thanksB.A.Bernevigfordiscussionandfeedback,andthanksHuanHeforthefirstintroductiontothestudyof quantummany-bodyscars.C.L.andJ.R.thankShuChenforhisencouragementandsupportattheearlystageofthisproject.C.F.andJ.R.acknowledgesupportfromMinistryofScienceandTechnologyofChinaunderGrantNo.2016YFA0302400,NationalScienceFoundationofChinaunderGrantNo.11674370,andChineseAcademyofSciencesunderGrantNo.XXH13506-202andXDB33000000.C.L.acknowledgessupportfromMinistryofScienceandTechnologyofChinaunderGrantNo.2016YFA0300600. *Theseauthorscontributedequallytothiswork.†cfang@[1]
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