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Many-BodyLocalization GeoffreyJi Outline • Aside:Quantumthermalization;ETH • Single-particle(Anderson)localization • Many-bodylocalization • Somephenomenology(l-bitmodel) • Numerics &Experiments Thermalization • ClassicThermalization: • Systemincontactwithreservoir– energy(andparticle)flow • Candescribesystembyfew parametersafterlongtime Classical Reservoir • P,T,V,S,mu,etc.… • Quantumsystems? • Alsocanuseclassicalreservoir • Butisconceptofreservoir necessary? • No! Quantum ?? Closed-SystemQuantumThermalization • Inquantumsystem,initialstate informationperfectlypreserved • Unitaryevolution • Systemattimetdependsonexact initialstate– nolossofinformation • Obviousineigenstatebasis • Directlycontradictsthermalization! • Butdoessubsystem lookthermal atlongtimes? • Canloseinformationaboutinitial statetoothersubsystems! SubsystemThermalization • Givenasubsystem!: • Needstobelocal – measurewith few-bodyoperators.E.g.compactin real-space • Givensomeglobalinitialstate": • musthavevanishingrelative uncertaintyinE,m,N,etc.,asthe reservoir!̅ → ∞ size(nocatstates) • Reduceddensitymatrixisthermal atlongtimesforlargesystem! lim ", - = "/ (1) )→*,,̅→* A.M.Kaufman,et.al.Science353,6301(2016) EigenstateThermalization Hypothesis(ETH) • Eigenstatesdon’ttimeevolve • ETHsaysallmany-bodyeigenstatesarealreadythermal(ETH) • Theexactstatementismoreprecise(equivalenceof • Sosubsystemthermalization happensbecauseeigenstatesalready arethermal! • Where’stheproof??(ofeithersubsystemthermalization orETH) • NONE (atleastingeneral) • Strongnumericalevidence(exactdiagonalization) BreakdownofThermalization • Somesystemsfailtothermalize ingeneral • Integrable systems • Conservedquantities~4 • Localizingsystems • Andersonlocalization– single particles • Many-bodylocalization (interactions) AndersonLocalization • Discussedbrieflyinclass • Reminder: • Freespinless fermionsindisorder • Quantuminterferencefrom coherentbackscattering • 5 ≤ 2:alwayslocalizing(i.e. wavefunction decayswithlength dependentondisorder) • 5 > 2:localizingtransitionfor somecriticaldisorderstrength AndersonLocalization • Anderson(1958):Perturbativeexpansioninpotential(not diagrammatics butessentiallythesameasinclass) • Coherentbackscatteringfromhigherorderexpansiontermscauses localization(divergenceofresistivity) • Localizationmeansfailuretothermalize! Whataboutinteractions? • Dotheymatter? • Nogeneralanswer! • Basko,Aleiner,Altshuler (2006): • “electron–electroninteractionalone cannotcausefiniteconductivityeven whentemperatureisfinite,butsmall enough”– MBL! • Perturbativeapproach • Mobilityedge(unboundsystems) • Imbrie (2014): • In1Dspinchains,non-perturbative effectsbounded– resonancesrare • Not100%foolproof IntuitivePicture(PerturbativeApproach) • Weakinteractionsshouldonly modifytheeigenstates! • Startfromlocalizedmodel,dress withinteractions • Newwavefunctions haveweak hybridization • Mustdocalculationtoseeif enoughtothermalize! E 9=0 9>0 Phenomenology • Ingeneral,localizingphasesofmodelswithinteractionsdifferfrom non-interactingones.Butstatementsdifficulttoprove. Thermal Single-ParticleLocalizing Many-BodyLocalizing Conducting Insulating Insulating Initial conditionshidden Some initialconditions persist Some initialconditions persist ETHholds ETH fails ETHfails Power-lawentanglement Noentanglement Logarithmicgrowthof entanglement Dephasing &dissipation Nodephasingor dissipation Dephasing(from entanglement),no dissipation R.Nandkishore,et.al.,Annu.Rev.Condens.MatterPhys.2015.6:15-38 PhenomenologicalModel • Ingeneral,hardtogetanalyticresults.HeavydependenceonED • Assumingthesingle-particlewavefunctions weaklyhybridize(i.e. localization),wecanconstructeffectiveHamiltonian! • ConsidersomeHamiltonianwith2-statelocalDOF(spinorcharge) andlocalinteractions,e.g.: • • • • Heisenbergw/randomz-field(alsoknownasrandom-fieldXXZ) Single-spinHubbardw/NNinteractionsandrandomoffsets TransverseIsing w/weak(integrability-breaking)x-coupling Kitaev model(mapstoabove) PhenomenologicalModel(Huse,2014) • LocalDOF:Np-bits(physicalbit),i.e.pseudospins • Strongdisorder:alleigenstateslocalized(finite#ofeigenstates) • SocanalsodefineNl-bits(localizedbit)thatarealsopseudospins • l-bitoperatorsare ;< ,andfornon-interactingcaselooklike, ;= ~ ∑ G ? @ AB CAD E FG • Ingeneral,ifeigenstatesarelocalized(notransport), ∞ I = J ℎ< L<M + J O<,P L<M LPM + J J Q<,P,{S} L<M LSM1 … LSMW LPM < <,P W=1 <,P,{S} Makingl-BitsPerturbatively • Couldgoperturbativeroute • Intuitively,givesusalltherightthings! • ;= composedofFG withexponentialfalloff • TroublewhenΔ=,=YG = 0!(resonances) • Somewhatresolvablebyarguingsufficientrarityofresonances– noteasy andnotgeneral(Imbrie 2014) • Formulationispainful(Basko 2005) • Let’snotuseperturbationtheory… Makingl-BitsNon-Perturbatively… • Huse (2014):InMBLphase,shouldbepossibletoconstruct{;= } • EachL=[ ] is(trivially)composedofsumsoftensorproductsof\= (includingidentityspinoperators) ] • Tensorproduct⊗=,] _=,] \= hasmaximumdistancebetweennonidentityoperators • Choose{;= } tominimizeaverageofaveragedistanceofproductsin;= • Havetodothissincepossiblethatsome;= looknon-local,butare exponentiallyrare • Resultshouldbefinite! PhenomenologicalModel * I = J ℎ= L=[ + J O=,G L=[ LG[ + J J Q=,G,{`} L=[ L`[a … L`[b LG[ = =,G cde =,G,{`} • For9 = 0,O=,G = Q=,G,{`} = 0 (nodephasing) • Ergodicitybreaking:alllocal L=[ areconstantsofmotion! • Long-rangeentanglementthroughphaseinformation Short-TimeGrowthofEntanglement • Thel-bitsareexponentiallylocalizedinrealspace(p-bits) • p-bitsinteractlocally– l-bitinteractionsuppressedexponentially vs. distance • Startinproductstateofl-bits • Shorttimedynamics:entanglementwithnearby l-bits(localizationlength) • “Area-law”entanglement– particleentanglementgrowsasthecoordination number,subsystementanglementproportionaltobordersize,notvolume • Area-lawentanglementgenerallytruefornon-interactingsystemsas well Long-TimeGrowthofEntanglement(NoMBL) • Thermalizingsystem: • Considerspin-chainlengthf,interactionsofO • Entanglementfromend-to-endshouldtakeg~hi • EssentiallyaLieb-Robinsonvelocity • Intuition:site< getsentangledwithsite< + 1,site< + 1 getsentangledwith site< + 2 andcauses< and< + 2 tobecomeentangled • Aside:Sites<,P entangledifyoutraceoversite< andsiteP isnowmixed • Localization,nointeractions: • Effectivel-bitHamiltonianisolatesl-bitsfromoneanother • Phasesofl-bitsevolveindependently Long-TimeGrowthofEntanglement(MBL) • Withouttransport,entanglementgrowthnotpower-law • Phaseof;= dependsonLG[ ,not;G ! • l-bitscannottransferentanglementamongstoneanother • EntanglementonlythroughdirecttermsinHamiltonian • So,rateofentanglementgrowthgoeslike: * jkk O=,G = O=,G + J J Q=,G,{`} L`[a … L`[b cde {`} • p-bitsexponentiallylocalinl-bits=>couplingsdieoffexponentially=> entanglementgrowslogarithmically Numerics • Muchofpreviousdiscussionnotproven– neednumerics toverify • DMRGonXX-modelwithstaticmagneticfieldandvariableOz interaction(Bardarson,2012),L=10(and20) • Startwithspinsrandomlyalignedinz-axis,quenchinteractionson • DMRGreducescomplexityofproblembytruncatingunnecessaryphasespace • Aligninginz-axisreducesentanglement,makingcomputationfeasible Numerics – Results • Saturationofentanglement entropyfornoOz (forsubsystem) • XX-modelmapsontononinteractingfermions • ForOm > 0,unbounded logarithmicgrowthin entanglemententropy • Evenforverysmallinteractions! • Universalbehavior • Eventuallysaturates(finite systemlimitsmaxentropy) Numerics – Results&Discussion • Observationofvolume-law entanglemententropygrowth • Butsaturatedentanglement entropypersitemuchlowerthan thermal • Perhapslimitedthermalization withrestrictedensemble • Particlenumberfluctuationsdo notshowphasetransition • Growthinentanglementis qualitiative differencebetween AndersonlocalizationandMBL! ExperimentalWork • MBLdifficulttoobserveinsolid-statesystem • Dependsonisolatedquantumsystem! • BathconnectiondestroysMBLsignatures(Johri 2015) • AdvancesinAMOhavemadeexperimentspossible • Trappedions(Smith2016) • Loggrowthofquantumfisherinformation(lowerboundofentanglemententropy) • Coldatoms(Schrieber 2015) • Futureexperiments? • Directmeasurementofentanglemententropypossible!(Islam2015) References • TheoreticalReviews: • Nandkishore,R.&Huse,D.A.Many-BodyLocalizationandThermalization inQuantumStatisticalMechanics.AnnualReview ofCondensedMatterPhysics6,15–38(2015). • TheoreticalResults: • Anderson,P.W.AbsenceofDiffusioninCertainRandomLattices.PhysicalReview109,1492–1505(1958). • Basko,D.M.,Aleiner,I.L.&Altshuler,B.L.Metal–insulatortransitioninaweaklyinteractingmany-electronsystemwith localizedsingle-particlestates.AnnalsofPhysics321,1126–1205(2006). • Imbrie,J.Z.OnMany-BodyLocalizationforQuantumSpinChains.JournalofStatisticalPhysics163,998–1048(2016). • PhenomenologicalModels: • Huse,D.A.,Nandkishore,R.&Oganesyan,V.Phenomenologyoffullymany-body-localizedsystems.PhysicalReviewB90, (2014). • Numerics: • Bardarson,J.H.,Pollmann,F.&Moore,J.E.UnboundedGrowthofEntanglementinModelsofMany-BodyLocalization. PhysicalReviewLetters109,(2012). • Experiments: • Smith,J.etal.Many-bodylocalizationinaquantumsimulatorwithprogrammablerandomdisorder.NaturePhysics12,907– 911(2016). • Schreiber,M.etal.Observationofmany-bodylocalizationofinteractingfermionsinaquasirandom opticallattice.Science 349,842–845(2015).