Download Many-Body Localization

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).