Inevitable Power-law Behavior of Isolated Many-Body Quantum Systems and How It Anticipates Thermalization

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by  Marco Távora

Inevitablepower-law behaviorofisolatedmany-body quantum systemsand how itanticipates
thermalization
Marco Ta´vora1, E. J. Torres-Herrera2, and Lea F. Santos1
1Department of Physics, Yeshiva University, New York, New York 10016, USA and
2Instituto de F´ısica, Universidad Auto´noma de Puebla,
Apartado Postal J-48, Puebla, Puebla, 72570, Mexico
(Dated:November3,2016)
6
1 Despite being ubiquitous, out-of-equilibrium quantum systems are much less understood than systems at
0 equilibrium. Progressinthefieldhasbenefitedfromasymbioticrelationshipbetweentheoreticalstudiesand
2 newexperimentsoncoherentdynamics. Thepresentworkstrengthensthisconnectionbyprovidingageneral
pictureof the relaxation process of isolatedlatticemany-body quantum systemsthat are routinelystudied in
v
experimentswithcoldatoms,ionstraps,andnuclearmagneticresonance.Weshownumericallyandanalytically
o
N thatthelong-timedecayoftheprobabilityforfindingthesysteminitsinitialstatenecessarilyshowsapower-law
behavior∝t−γ. Thishappensindependentlyofthedetailsofthesystem,suchasintegrability,levelrepulsion,

2 andthepresenceorabsenceofdisorder. Informationaboutthespectrum,thestructureoftheinitialstate,and
thenumberofparticlesthatinteractsimultaneouslyiscontainedinthevalueofγ. Fromit,wecananticipate

] whethertheinitialstatewillorwillnotthermalize.
h
c
e Introduction. A great deal of effort has recently been Ψ(0) with the eigenstates ψ of H, and ρ (E) =
m | i | αi 0
putinto improvingourunderstandingof isolatedmany-body C(0) 2δ(E E ) is the energy distribution of Ψ(0)
t- quantum systems quenched far from equilibrium. This is in Pweαig|hteαd|bythe−compαonents C(0) 2,theso-calledloca|ldeni-
a partmotivatedbythepossibilityofinvestigatingthecoherent | α |
sityofstates(LDOS).Thesurvivalprobabilityistheabsolute
t
s evolution of these systems for long times with different ex- squareoftheFouriertransformoftheLDOS.Allinformation
. perimental setups, including those with ultracold atoms [1],
t abouttheevolutionofF(t)iscontainedinρ (E).
a trapped ions [2, 3], and nuclear magnetic resonance [4, 5]. 0
m We verified that the initial decay of the survival probabil-
Alignedwith these efforts, thiswork characterizesand justi-
ity is dissociated from the regime (integrable or chaotic) of
- fiesthedynamicalbehavioratdifferenttimescalesofexperi-
d the Hamiltonian [18–23], but dependson the strength of the
mentallyaccessibleintegrableandchaoticlatticemany-body
n perturbation. We now show that at long times, regardlessof
quantumsystemswithandwithoutdisorder. Fromthisanal-
o
howfasttheinitialevolutionmaybe,thedynamicsnecessar-
c ysis,anewcriterion,basedexclusivelyondynamics,isintro-
ily slows down and becomes power-law, F(t) t γ. The
[ ducedforidentifyingwhichsystemscanthermalize. ∝ −
characterizationofthelong-timedynamicsanditsconnection
Thesurvivalprobability(probabilityforfindingthesystem
4 with the viability of thermalization are the central topics of
v initsinitialstateattimet)andtheLoschmidtecho(measure
thiswork.
7 oftherevivaloftheinitialstateafteratime-reversaloperation)
We show that in realistic lattice many-bodyquantum sys-
0 have been extensively considered in the analysis of out-of-
8 equilibriumquantumsystems [6–13]. Severalworkstried to tems with two-body interactions, 0 γ 2. The value of
≤ ≤
5 establishacorrespondencebetweentheinitialexponentialor the power-law exponentindicates the level of delocalization
0 of the initial state in the energy eigenbasis. When the ini-
Gaussian decayswith quantumchaos[12–17] andothersfo-
1. cusedontheonsetofpower-lawdecaysatlongtimes[6–11]. tialstate samplesonlya portionof the Hilbertspace and the
0 In the case of continuous models, the algebraic behavior of LDOS is sparse, γ < 1 and thermalization is not expected.
6 thesurvivalprobabilityhasbeenassociatedwiththepresence Whentheinitialstate ischaotic, sothatitscomponentsCα(0)
1
ofboundsinthespectrum[6–8]whileindisorderednoninter- are uncorrelatedand spread over its entire energyshell [24–
:
v actingsystemsatthemetal-insulatortransition,thepower-law 27],thermalizationshouldoccur[27–33]. Inparticular,when
i exponenthasbeenrelatedwithfractaldimensions[9–11].Ex- theLDOSisergodicallyfilled,thenγ =2.Fromthevaluesof
X
changesbetweenthesedifferentcommunitieshavebeenvery γ,onecanthusanticipatewhethertheinitialstatewillorwill
r
a limited. Here,weunifythesemultipleperspectivesintoasin- notthermalize. We also discuss the non-realisticscenarioof
gleframeworkanduseittodescribetheevolutionofthesur- fullrandommatrices,wherethepower-lawexponentreaches
vivalprobabilityoflatticemany-bodyquantumsystems. theupperboundγ =3.
The survival probability (or fidelity) of the initial state is Time scales. The system is initially prepared in an eigen-
definedas state of the unperturbed Hamiltonian H , which is abruptly
0
F(t) ≡ hΨ(0)|e−iHt|Ψ(0)i 2 qoufetnhcehepdertiunrtboatHion=V.HA0t+vegrVy,shwohrterteimgesi,sththeedsetcreanygothf
(cid:12)(cid:12) 2(cid:12)(cid:12) 2 the survival probability is quadratic [34], as derived from
wHh,eCreα(0E)α==ar(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)eψXααthΨe|C(eα0(i0)g)e|2naevr−eailEutheαest(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)oovf=etrhl(cid:12)(cid:12)(cid:12)(cid:12)aZepdssyEosefte−mtihEetHρina0mi(tEiial)lto(cid:12)(cid:12)(cid:12)(cid:12)sntia(a1tne) [EtiPhn0eitαi=ael|xCspPtα(aa0ntα)es|.|i2Co(nEα(0αF)|2−(Et Eα≪0=)2]hσ1Ψ0/−2(10)i)s|H≈th|eΨ1w(0−id)ithσis02ottf2h,ethweenhLeerDrgeOyσSo0fatnh=de
h | i
2
Aftertheinitialquadraticbehavior,whetherF(t)switches ables and IPR 3/ [28]. Even though realistic chaotic
0
∼ D
or not to an exponential decay depends on the strength g of many-bodyquantumsystems with two-bodyinteractionsare
theperturbation. TheexponentialdecayisvalidintheFermi not described by FRMs, because their Hamiltonian matrices
goldenruleregime,wherethetypicalmatrixelementsofgV arebanded,sparse,andrandomelementsmaynotevenexist,
are larger than the mean level spacing and the LDOS has a theystillfollowrandommatrixstatisticsandtheirbulkeigen-
Lorentzianform.However,forverystrongperturbations,g statesareclosetorandomvectors[28,49,50]. Afterastrong
→
1,theLDOSisbroader.Inmany-bodyquantumsystemswith perturbationinto such Hamiltonians, initial states with ener-
two-body interactions, where the density of states is Gaus- giesawayfromtheedgesofthespectrumalsogiveveryfilled
sian [35–37], the limiting shape of the LDOS is also Gaus- LDOS[24–32].
sian, resulting in the Gaussian decay F(t) = exp( σ2t2) Case 1(i) holds for realistic chaotic many-body quantum
− 0
[14, 15, 18–23, 38]. Exponential and Gaussian decays can systems,wheretheLDOSisGaussian,whichleadstoγ =2.
thusoccurinbothintegrableandchaoticmodels[18–23].The ForFRM,theLDOSisasemicircle[18],socase1(ii)applies
picturebecomesmoresubtleatlongtimes,wherethepower- andγ =3.
lawbehavior t γemergesandthefillingoftheLDOSplays Incase2,thepower-lawexponentisobtainedfromthecor-
−
∝
akeyrole. relationfunctionC(ω) C(0) 2 C(0) 2δ(E E ω)
Causes of the power-law decay. We discuss two distinc- ≡ α,β| β | | α | α− β−
presentin F(t) = ∞ dωPeiωtC(ω). A power-lawdecayof
tive causes for the long-time algebraic decay of the survival
C(ω 0) ωγ R1−,∞with γ < 1, leads to F(t) t γ [9–
probability. → ∝ − ∝ −
11, 51–53]. The more correlated the componentsof Ψ(0) ,
Case1 is related to the unavoidable presence of a lower | i
the smaller the exponent γ. This exponent coincides with
boundE intheenergyspectrumofanyrealquantumsys-
low thefractaldimensionφobtainedfromthescalinganalysisof
tem. This point was put forward already in 1958 [39] and
IPR φ. ThisrelationwasfoundinstudiesofAnderson
in several other early works[40–45]. At long times, the en- 0 ∝D−
localization [9–11] and of many-body localization [22, 23].
ergy bound leads to the partial reconstruction of the initial
We show thatit holdsalso innoninteractingintegrablemod-
state.Thisresultsinthepower-lawdecayofcontinuousmany-
els.
particle models [6–8] and, as explained here, also of finite
This work analyzes how γ depends on the properties of
lattice many-body quantum systems with ergodically filled
the spectrum, the structure of the initial state, and the num-
LDOS.
ber of particles that interactsimultaneously. We consider fi-
Case2 is induced by the correlations that are present in
nite many-body quantum systems described by realistic lat-
nonchaotic eigenstates. They are typical of disordered sys-
ticemodelswithtwo-bodyinteractionsandbybandedrandom
tems undergoinglocalization with [22, 23] or without inter-
matrices. All accessible power-law exponents are reached
actions[9–11]and,asarguedhere,appearalsoincleaninte-
with the disorderedmodels, while with the clean Hamiltoni-
grablesystems. The power-lawexponentdueto correlations
ans,westudysomespecificvalues.
issmallerthanthatresultingfromenergybounds.
Realistic many-body quantum systems. We consider one-
The exponents of case 1 can be derived from asymptotic
dimensional spin-1/2 models with L sites described by the
expansions of the integral form of Eq. (1), assuming that
followingHamiltonian,
ρ (E)isabsolutelyintegrable[46]andthatitsderivativesex-
0
istandarecontinuousin[E , ]. Twoscenariosareidenti-
low L
∞
fied[47,48]: H = h Sz +H +λH , (2)
((iii))IfIflimρE(→EE)lodwecρa0y(Es )ab>ru0p,tltyhecnloastelotnogtthimeelsowFe(rt)b∝out−nd2., nX=1 n n NN NNN
0 H =J SxSx +SySy +∆SzSz ,
such that ρ (E) = (E E )ξη(E) with 0 < ξ < 1 and NN n n+1 n n+1 n n+1
lim 0η(E)>0,th−enFlo(wt) t 2(ξ+1). Xn (cid:0) (cid:1)
TEh→esEelorwesultshavebeenobtain∝edf−orcontinuousfunctions. HNNN = J SnxSnx+2+SnySny+2+∆SnzSnz+2 .
Yetweshowthattheyremainvalideveninthecaseofdiscrete Xn (cid:0) (cid:1)
spectra provided Ψ(0) is chaotic and the LDOS is ergodi-
| i It contains nearest-neighbor (NN) and possibly also next-
callyfilled.
nearest-neighbor(NNN)couplings;~ = 1andSx,y,z arethe
To determine if the initial state is chaotic, one performs n
spinoperatorsonsiten. h arerandomnumbersfromauni-
scaling analysis of the inverse participation ratio (IPR) of n
form distribution [ h,h]; the system is clean when h = 0
Ψ(0) writtenintheenergyeigenbasis,IPR C(0) 4. −
| i 0 ≡ α| α | and disordered otherwise. J is the coupling strength, ∆ the
IPR−01istheeffectivenumberofenergyeigenstatePscontribut- anisotropyparameter,andλ theratiobetweenNNNandNN
ingtotheinitialstate. Achaotic Ψ(0) samplesmostenergy couplings. J = 1 sets the energy scale. The Hamiltonian
| i
eigenbasiswithoutanybias,soIPR0 −1,where isthe conservestotalspininthez direction z. We workwith the
dimension of the Hilbert space. Hen∝ceD, as the systDem size largestsubspace z =0ofdimensionS =L!/(L/2)!2.
increases,ρ0(E)becomeshomogeneouslyfilledandcloseto TheintegrableSlimitsofH includethDecleannoninteracting
an absolutely integrablefunction. An illustrative example is XX (∆,λ,h = 0)and theclean interactingXXZ (∆ = 0,
thatofanarbitraryinitialstateprojectedontotheeigenstates λ,h=0)models.Thesystembecomeschaoticasλincre6ases
of a full random matrix (FRM). Since these eigenstates are fromzero[54–57]andthelevelspacingdistributionchanges
pseudo-random vectors, the overlaps C(0) are random vari- from Poisson [58] to a Wigner-Dysonform [59]. It also be-
α
3
caonmdhes<chJao[t6ic0–w6h2e]n. thedisorderstrengthincreasesfromzero F(t) = e−4Nσ022t2 (cid:12)erf(cid:16)E0−E√l2owσ0+iσ02t(cid:17)−erf(cid:16)E0−√Eu2pσ+0iσ02t(cid:17)(cid:12)2,
whereerfisthe(cid:12)errorfunctionand isanormalizationco(cid:12)n-
The initial states considered are site-basis vectors, where (cid:12) N (cid:12)
stant that depends on L through the energy bounds and σ .
the spin on each site either points up or down in the z- 0
At long times, after dropping the oscillations from the si-
direction. An example is the experimentally [63] accessible
nusoidal term cos[t(E + E )], the expression becomes
Ne´elstate, ... ,thathasbeenextensivelyusedin up low
studiesofth|e↑↓d↑y↓n↑am↓↑i↓csofiintegrablespinsystems. Site-basis F(t ≫ σ0−1) ≃ (2πσ02t2N2)−1 k=up,lowe−(Ek−E0)2/σ02,
vectorsevolveunderH (2)afterastrongperturbation,where fromwherethet−2power-lawdecPayisevident.
the anisotropyparameter is quenchedfrom ∆ to a fi- When h = 1, we get γ 1. This curve is depicted with
nitevalue. TheenvelopeoftheLDOSforthese→in∞itialstates athicklinein Fig.1(a). A∼bovethisline, h > 1andγ < 1.
isthereforeGaussian. Anexamplewithγ 1/2isisolatedinFig.1(b). Thisγ is
∼
closetotheexponentφobtainedfromthescalinganalysisof
Realistic disordered systems. Figure 1 shows the survival
IPR φ [22]. Thisexamplebelongstocase2.
probability of site-basis vectors evolving under H (2) with 0 −
∝D
Figure 1 (a) demonstrates that with the disordered XXZ
∆ = 1, λ = 0 and various values of h. The initial decay
model, we can obtain all power-law exponentsaccessible to
is Gaussian, as expectedfromthe Gaussian LDOS. Itagrees
verywellwiththeanalyticalexpressionF(t)=exp( σ2t2), realistic lattice many-body quantum systems with two-body
− 0 interactions.Byvaryingh,everyγ [0,2]canbereached.
as seen for the bottom curve of Fig. 1 (a). Subsequently
Bandedrandommatrices. Algebr∈aicdecaysfasterthant 2
the dynamics slows down and becomes a power-law for all −
alsosignaltheergodicfillingoftheLDOS.Theyarepossible
curves.
ifinsteadoftwo-bodyinteractions,many-bodyrandominter-
actionsareincluded. Asthenumberofparticlesthatinteract
100
(a) 0.3 (b) simultaneouslygrows,increasingthenumberofuncorrelated
γ ∼1/2 nonzero elements in the Hamiltonian matrix, the density of
>0.2
10-1 <f statestransitionsfromGaussiantoasemicircle[35]. Thelat-
0.1 ter is typical of FRMs [59]. This transition is reflected also
h=1.75
> 0 in the shape of the LDOS [18, 19, 26, 67, 68]. The Fourier
<F10-2 0.60 20 40 60 transformof a semicircle gives F(t) = [ (2σ t)]2/(σ2t2),
(c) J1 0 0
where is the Bessel function of the first kind [18, 19].
0.4 J1
10-3 <f>0.2 γ ∼ 2 Tashyemdpetoctaiyceaxtpsahnosirotntirmeveesalissafapsotwerert-hlaawn dGeacuasysiwanithaγnd=th3e,
10-4 1 100 00 5 h=01.20 aFm(tple≫ofσ0−ca1s)e≃1(i[i1),−wshiner(e4σf0otr)]t/h(e2πseσm03ti3c)ir.clTe,hiξs is=an1e/x2-,
t t
η(E)=(2πσ2) 1(2σ E)1/2,andE = 2σ .
0 − 0− low − 0
FIG.1: Survivalprobability(a)andf(t)(b),(c). In(a): frombot-
tomtotop,h = 0.2,0.3,...0.9,h = 0.95,1,1.25,...3,h = 3.5. 0
10
Thicksolidline: h = 1withγ ∼ 1. Circles: analyticalGaussian
decayF(t)=exp(−σ02t2).In(b)and(c):Numericalcurve(solid),
const−L−1lnt−γ (dashed). Averagesover105dataofdisorderre- -1
10
alizationsandinitialstateswithE0∼0;L=16,closedboundaries.
>
F
< -2
10
When the disorderstrength is small, 0 < h < 1, the sys-
tem is chaotic and the LDOS is very filled. This is corrob-
oratedfrom the analysisof levelstatistics and by computing 10-3
theinverseparticipationratioaveragedoverinitialstatesand
-2 -1 0 1 2
randomrealizations. One findsthat IPR 1. For the 10 10 10 10 10
h 0i ∝ D− t
value of h where IPR is maximum, the decay of F(t) at
0
long times is th2, asiillustrated with the bottom curve in
∝ − FIG. 2: Survival probability for basis vectors evolving under
Fig.1(a).Forothervaluesofhin(0,1],wehavetheinterme-
PBRM with b = 0.1,0.5,1,2,5,10,20,50,100,3000 (solid)
diateregion,where1 γ <2. Thesevaluesmayresultfrom from top to bottom. They correspond respectively to the fitted
≤
acompetitionbetweenweakcorrelationsandenergybounds, γ ∼ 0.1,0.5,0.6,0.7,0.9,1.2,1.4,1.9,2.2,2.8 (dashed). Analyt-
butthisneedstobefurtherinvestigated. ical F(t) = [J1(2σ0t)]2/(σ02t2) (dotted); D = 3432. Averages
The bottom curve of Fig. 1(a) is isolated in Fig. 1(c), over100realizationsand343initialstateswithE0∼0.
which shows the rescaled survival probability f(t) =
(1/L)lnF(t) [64, 65]. For L 1, this quantity is To illustrate the increase of the value of γ from 2 to the
− ≫
independent of L [66]. Figure 1 (c) is a clear exam- upper bound γ = 3, we consider power-law banded ran-
ple of the power-law decay caused by energy bounds [case dommatrices(PBRM)[69–71]. DespitethesuccessofFRMs
1(i)]. The Fourier transform of a Gaussian LDOS that has in describing statistically the spectrum of complex systems,
lower E and upper E bounds, as in our case, leads to they imply the unphysical scenario of all particles interact-
low up
4
ing simultaneously. Banded random matrices were intro- valuesofλand∆, including∆ = 0;andotherinitialstates.
duced [67] in an effort to better approach random matrices A t 2 decay has also been speculated for the chaotic Ising
−
to real systems. We use PBRMs that preserve time rever- modelwithlongitudinalandtransversefields[75].
sal symmetry and whose elements are real random numbers An analyticalexpressionexistsforF(t) for the Ne´elstate
frHom2 a G=au1s/s[i1an+di(sntribumtio)n/b[722]]f:orhHnn=nim=. 0T,hheHvna2nluie=of2b, epvaonlsvioinngfuonrdleorngthetimpeersi,odLict X1/X2 mod0e,l g[6iv5e,s7f6]N.e´el(Itt)s ex-
dhetnermmiineshowfas|tthe−elemen|tsdecrea6seastheymoveaway L 1ln 2 L 1+2 1Lt 1−/2 , →as indeed confiXrmXed wi→th
− − − −
fromthediagonal.Whenb ,thePBRMcoincideswitha t−he dashe(cid:2)d line(cid:0)in Fig. 3(d). Su(cid:1)c(cid:3)h small γ indicates that the
→D
FRM. LDOS is not ergodicallyfilled, as seen in Fig. 3(c) and cor-
InFig.2,weshowthesurvivalprobabilityforPBRMswith roboratedbelowbycalculatingIPR .
0
different values of b. As b grows from 50 to and the Amongthetotal = L!/(L/2)!2componentsoftheNe´el
∼ D
LDOStransitionsfromcase1(i)tocase1(ii),γincreasesfrom D
state,only2L/2arenonzeroandtheyareallequal, C(0) 2 =
2 to 3. In the other direction, as b decreases below 50, the 2 L/2 [76]. ThismeansthatIPR = 2 L/2. Using| thαe|Stir-
eigenstatesbecomeless spreadoutandγ decreasesbelow2. − 0 −
ling approximationfor large L, we have that ln Lln2.
With PBRMs, weobtaina generalpictureofthebehaviorof From lnIPR vs ln , we find that IPR D1/2≃, so φ =
thesurvivalprobability,coveringallvaluesofγ,withoutany 0 D 0 ≃ D−
1/2. One sees that, similarly to what is done in disordered
restrictiontoaspecificmodel.
systems [22, 23], the power-law exponent for the Ne´el state
Realistic cleansystems. InFig. 3, we studytheNe´elstate
in the XX model, γ = 1/2, can also be extracted from the
evolving under a clean chaotic Hamiltonian [Figs. 3(a) and
scalinganalysisofIPR .
3(b)] and under the XX Hamiltonian [Figs. 3(c) and 3(d)]. 0
Theenvelopeofthe LDOSisGaussianinbothcases(a)and The nonzero Cα(0) 2 are spread out in energy, result-
| |
(c),butvisiblysparseinFig.3(c). ing in a very sparse and inhomogeneous LDOS. The non-
ergodicity of this state indicates that thermalization should
0.5 not occur. One way to confirm thermalization is by
(a) 1 (b)
0.4 verifying the coincidence of the diagonal entropy Sd =
LDOS00..23 f0.5 γ∼2 −StPh =α|Clnα(0)|2αlen−|ECαα(/0T)|2−[7(7]aαnEdαthee−tEhαe/rTm)o/d(yTnamαicee−nEtrαo/pTy),
0.1 [29]. HerPe, Sd = (L/2)Pln2 and Sth = lnP[Note that
D
the Ne´el state has E = 0 and thus infinite temperature T].
0 0 0
-5 E0 5 0 5 1t0 15 20 Thetwoentropiesdonotcoincideeveninthethermodynamic
0.5 1.5 limit,where(Sth Sd)/L=ln√2.
(c) (d) −
0.4 Conclusions. We have shown that the long-time decay of
OS0.3 1 the survival probability in isolated lattice many-body quan-
D f tum systems is algebraic, F(t) t γ, be the system inte-
L0.2 0.5 ∝ −
γ∼1/2 grableor chaotic, interactingor noninteracting,clean or dis-
0.1
ordered. The entire range of γ [0,3] can be reached with
0 0 ∈
-5 0 5 0 5 10 15 bandedrandommatrices,whileforrealisticsystemswithtwo-
E t
body interactions, γ [0,2]. From the value of γ, we infer
∈
howmuchdelocalizedtheinitialstateisintheenergyeigen-
FIG. 3: LDOS [(a),(c)] and f(t) [(b),(d)] for the Ne´el state under
basis. Thisprovidesawaytoidentifywhethertheinitialstate
thechaoticopenH (2)withh = 0,∆ = 1/2,λ = 1[(a),(b)]and
undertheclosedXXH[(c),(d)].(a),(c):NumericalLDOS(shaded willthermalizebasedexclusivelyonitsdynamics.Exponents
area)andGaussianenvelope(solidline). (b): Numericalresultsfor γ 2signalergodicityandthereforethermalization. Advan-
L=22(light),L=24(dark),andconst−L−1lnt−2(dashed).(d): tag≥esofthisapproachtotheproblemofthermalizationinclude
L=400(solid)andfNe´el(t)(dashed). the following: any initial state can be considered, numerical
XX
methodsotherthanexactdiagonalizationareavailableforan-
In Fig. 3(b), we observe a power-law decay t 2. The alyzingdynamics,andanaturalconnectionisestablishedwith
−
agreement between the t 2 decay (dashed line)∝and our nu- experimentsthatroutinelystudythedynamicsofmany-body
−
merical results (solid lines) suggests that the LDOS must be quantumsystems.
ergodically filled and that thermalization should occur. In- Acknowledgments. This work was supported by the NSF
deed,theinverseparticipationratiooftheNe´elstateinFig.3 Grant No. DMR-1147430 and Yeshiva University. EJTH
(a)givesIPR 1andseveralstudiesforthismodelcon- acknowledges funding from CONACyT, PRODEP-SEP and
0 −
∝D
firmthermalization[29,30,73,74]. Wefoundγ =2alsofor VIEP-BUAP,Mexico.WethankAdolfodelCampo,Yevgeny
periodic boundary conditions; chaotic models with different BarLev,andMarcosRigolforusefuldiscussions.
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