Many-Body Localization in a Quasiperiodic System

Shankar Iyer and Gil RefaelDepartment of Physics, California Institute of Technology,

MC 149-33, 1200 E. California Blvd., Pasadena, CA 91125

Vadim OganesyanDepartment of Engineering Science and Physics,

College of Staten Island, CUNY, Staten Island, NY 10314The Graduate Center, CUNY, 365 5th Ave., New York, NY, 10016 and

KITP, UCSB, Santa Barbara, CA 93106-4030

David A. HuseDepartment of Physics, Princeton University, Princeton, NJ 08544

(Dated: April 15, 2013)

Recent theoretical and numerical evidence suggests that localization can survive in disorderedmany-body systems with very high energy density, provided that interactions are sufficiently weak.Stronger interactions can destroy localization, leading to a so-called many-body localization transi-tion. This dynamical phase transition is relevant to questions of thermalization in extended quantumsystems far from the zero-temperature limit. It separates a many-body localized phase, in whichlocalization prevents transport and thermalization, from a conducting (“ergodic”) phase in whichthe usual assumptions of quantum statistical mechanics hold. Here, we present numerical evidencethat many-body localization also occurs in models without disorder but rather a quasiperiodic po-tential. In one dimension, these systems already have a single-particle localization transition, andwe show that this transition becomes a many-body localization transition upon the introductionof interactions. We also comment on possible relevance of our results to experimental studies ofmany-body dynamics of cold atoms and non-linear light in quasiperiodic potentials.

I. INTRODUCTION

In one-dimensional systems of non-interacting parti-cles, an arbitrarily weak disordered potential genericallylocalizes all quantum eigenstates1,2. Such a system isalways an insulator, with a vanishing conductivity inthe thermodynamic limit. The question of how this pic-ture is modified by interactions remained unclear in thedecades following Anderson’s original work on localiza-tion3,4. Relatively recently, Basko, Aleiner, and Alt-shuler have argued that an interacting many-body systemcan undergo a so-called many-body localization (MBL)transition in the presence of quenched disorder. At lowenergy density and/or strong disorder, interactions areinsufficient to thermalize the system, so the system re-mains a “perfect” insulator (i.e. with zero DC conduc-tivity despited being excited); at higher energy densityand/or weaker disorder, the conductivity can becomenonzero and the system thermalizes, leading to a con-ducting phase5,6.

The MBL transition is rather unique for several rea-sons. First, in contrast to more conventional quantumphase transitions7, this is not a transition in the groundstate. Instead, the MBL transition involves the local-ization of highly excited states of a many-body system,with finite energy density. This means that the transi-tion differs from most metal-insulator transitions, whichare sharp only at zero temperature8. Furthermore, thisMBL transition is of fundamental interest in the contextof statistical mechanics. Local subsystems of interacting,

many-body systems are generically expected to equili-brate with their surroundings, with statistical propertiesof these subsystems reaching thermal values after suffi-cient time. Studies of how this occurs in quantum sys-tems have led to the so-called eigenstate thermalizationhypothesis (ETH), which states that individual eigen-states of the interacting quantum system already encodethermal distributions of local quantities9,10. However,the many-body localized phase provides an example of asituation in which the ETH is false, and the ergodic hy-pothesis of quantum statistical mechanics is violated11,12.Since the work of Basko et al., these intriguing aspects ofMBL have motivated many studies aimed at locating andunderstanding this transition in disordered systems11–23.

On the other hand, it is important to note that single-particle localization does not require disorder. In 1980,Aubry and André studied a 1D single-particle tight-binding model that omits disorder in favor of a potentialthat is periodic, but with a period that is incommensu-rate with the underlying lattice24. Harper had studieda similar model much earlier, but he had focused on aspecial ratio of hopping to potential strength25. Aubryand André showed that this point actually lies at a local-ization transition. It separates a weak potential phase,where all single-particle eigenstates are extended, froma strong potential phase, where all eigenstates are lo-calized. In the 1980s and 1990s, physicists continuedto study this quasiperiodic localization transition for itsown peculiarities and because it mimics the situation indisordered systems in d ≥ 3, where there is also a single-

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particle localization transition26–33. The AA model wasalso actively investigated in the mathematical physics lit-erature, because it involves a Schrödinger operator (i.e.the “almost Mathieu” operator) with particularly richspectral properties. The contributions of mathematicalphysicists put the initial work on Aubry and André onmore rigorous footing34–37. More recently, the AA modelhas been directly experimentally realized in cold atomexperiments38,39 and also in photonic waveguides40. Thepossibility of engineering quasiperiodic systems in thelaboratory has inspired new theoretical and numericalwork aimed at understanding the localization propertiesof such systems and how they differ from those with truedisorder41–47.

1. Statement of the problem and summary of the results

In this paper, we ask whether there can be a MBLtransition in an interacting extension of the AA model.More concretely, suppose we begin with a half-filled, one-dimensional system of fermions or hardcore bosons in aparticular randomly chosen many-body Fock state, withsome sites occupied and others empty. Such a configu-ration of particles is typically far from the ground stateof the system. Instead, by sampling the initial configu-ration uniformly at random (i.e. without regard to itsenergy content), we are actually working in the so-calledinfinite temperature limit. If the particles are allowed tohop and interact for a sufficiently long time, the standardexpectation is that the system should thermalize: that is,all microscopic states that are consistent with conserva-tion laws should become equally likely and local observ-ables should thereby assume some thermal distribution48.Can this expectation be violated in the presence of aquasiperiodic potential? In other words, can the systemfail to serve as a good heat bath for itself? If so, can thisbe traced to the persistence of localization even in thepresence of interactions?

The answer to both of these questions appears to be“yes.” We use numerical simulations of unitary evolutionof a many-body quasiperiodic system to measure threekinds of observables in the limit of very late times: thecorrelation between the initial and time-evolved particledensity profiles, the many-body participation ratio, andthe Rényi entropy. Our observations are consistent withthe existence of two phases in the parameter space of ourmodel that differ qualitatively in ergodicity. At finite in-terparticle interaction strength u and large hopping g,there exists a phase in which the usual assumptions ofstatistical mechanics appear to hold. The initial stateevolves into a superposition of a finite fraction of the totalnumber of possible configurations, and consequently, lo-cal observables approximately assume their thermal val-ues. This is the many-body ergodic phase. However, atsmall hopping g, there is a phase in which particle trans-port away from the initial configuration is not stronglyenhanced by interactions. The system explores only an

g

u

Many%body)ergodic)

Many%body))localized)

AA)localized) AA)extended)

FIG. 1: The proposed phase diagram of our interactingAubry-André model at high energy density. Interactions con-vert the localized and extended phases of the AA model intomany-body localized and ergodic phases and induce an expan-sion of the many-body ergodic phase. The phases of the in-teracting model differ qualitatively from their non-interactingcounterparts. The differences are explained in Section IV be-low.

exponentially small fraction of configuration space, andlocal observables do not even approximately thermal-ize. This is the many-body localized phase. Figure1 presents a schematic illustration of the proposed phasediagram. Although interactions induce an expansion ofthe ergodic regime, the localized phase survives at finiteu, and consequently, there is evidence for a quasiperiodicMBL transition58.

There has certainly been substantial previous work onlocalization in many-body quasiperiodic systems. Forinstance, Vidal et al.33 adapted the approach of Gia-marchi and Schulz49 to study the effects of a perturbativequasiperiodic potential on the low-energy physics of in-teracting fermions in one dimension. Very recently, Heet al.45 studied the ground state Bose glass to super-fluid transition for hardcore bosons in a 1D quasiperiodiclattice. Our work differs fundamentally from these andmany other studies precisely because it focuses on non-equilibrium behavior in the high-energy (infinite temper-ature) limit and argues that a localization transition caneven occur in this regime.

2. Organization of the paper

We begin our study in Section II by introducing ourinteracting extension of the standard AA model. Sincethe MBL transition is a non-equilibrium phase transi-tion, our goal is to follow the real-time dynamics. Tosimplify this task, we describe a method of modifyingthe dynamics of our model, such that numerical integra-tion of the new dynamics is somewhat easier than the

3

original problem. In Section III, we introduce the quan-tities that we measure in our simulations and present thenumerical results. Then, in Section IV, we argue thatour data points to the existence of many-body localizedand many-body ergodic phases by proposing model late-time states for each of these regimes and comparing tothe numerical results from Section III. Next, in SectionV, we extract estimates for the phase boundary from ourdata, motivating the phase diagram in Figure 1. Finally,we conclude in Section VI by summarizing our results,drawing connections to theory and experiment, and sug-gesting avenues for future extensions of our work.

We relegate two exact diagonalization studies to Ap-pendix A. First, we examine the impact of our modifieddynamics upon the single-particle and many-body prob-lems. Second, we study the many-body level statistics ofthe interacting model. We find evidence for a crossoverbetween Poisson and Wigner-Dyson statistics, consistentwith the usual expectation in the presence of a localiza-tion transition50.

II. MODEL AND METHODOLOGY

In this section, we motivate and introduce our modeland our numerical methodology for studying real-timedynamics.

A. The “Parent” Model

We would like to consider one-dimensional lattice mod-els of the following general form:

Ĥ =

L−1∑j=0

[hj n̂j + J(ĉ

†j ĉj+1 + ĉ

†j+1ĉj) + V n̂j n̂j+1

](1)

Here, ĉj is a fermion annihilation operator, and n̂j ≡ ĉ†j ĉjis the corresponding fermion number operator. The threeterms in the Hamiltonian (1) then correspond to an on-site potential, nearest-neighbor hopping, and nearest-neighbor interaction respectively. For now, we leave theboundary conditions unspecified. In 1D, the Hamiltoni-ans (1) for hardcore bosons and fermions differ only in thematrix elements describing hopping over the boundary.With open boundary conditions, the Hamiltonians (andconsequently all properties of the spectra) are identical.

If we set V = 0 in the Hamiltonian (1) and take hj tobe genuinely disordered, we recover the non-interactingAnderson Hamiltonian. If we then turn on a finite V = J ,we obtain a model that is related to the spin models thathave been studied in the context of MBL12,19. Alterna-tively, suppose we set V = 0 again and take:

hj = h cos(2πkj + δ) (2)

With a generic irrational wavenumber k and an arbitraryoffset δ, we obtain the non-interacting AA model24. For

our purposes, we would like to use an incommensuratepotential of the form (2), with h = 1 and g ≡ Jh and u ≡Vh left as tuning parameters to explore different phasesof the model (1).

Before proceeding, we should briefly review what isknown about the single-particle AA model. With peri-odic boundary conditions and δ = 0, this model is self-dual24,41. The self-duality can be realized by switchingto Fourier space (cj =

1√L

∑q e

iqjcq) and then perform-

ing a rearrangement of the wavenumbers q such that thereal-space potential term looks like a nearest-neighborhopping in Fourier space and vice versa. On a finite lat-tice of length L with periodic boundary conditions, sucha rearrangement is possible whenever the wavenumberof the potential k = `L such that ` and L are mutuallyprime. The duality construction reveals that, if the AAmodel has a transition, it must occur at g = 12 . In thethermodynamic limit, there is indeed a transition at thisvalue for nearly all irrational wavenumbers k26. Wheng > 12 , all single-particle eigenstates are spatially ex-tended, and by duality, localized in momentum space;when g < 12 , all single-particle eigenstates are spatiallylocalized, and by duality, extended in momentum space.Exactly at g = 12 , the eigenstates are multifractal

31,32.The spatially extended phase of the AA model is charac-terized by ballistic, not diffusive, transport24. Recently,Albert and Leboeuf have argued that localization in theAA model is a fundamentally more classical phenomenonthan disorder-induced Anderson localization, and thatthe AA transition at g = 12 is most simply viewed asthe classical trapping that occurs when the maximumeigenvalue of the kinetic (or hopping) term crosses theamplitude of the incommensurate potential41.

B. Numerical Methodology and Modification ofthe Quantum Dynamics

Probing the MBL transition necessarily involves study-ing highly excited states of the system, and this precludesthe application of much of the extensive machinery thathas been developed for investigating low-energy physics.Consequently, several studies of MBL have resorted toexact diagonalization or other methods involving similarnumerical cost11,12,16. We too use a numerical method-ology that scales exponentially in the size of the system.However, in order to access longer evolution times inlarger lattices, we introduce a modification of the quan-tum dynamics. This modification is inspired by a schemeused previously by two of us in a study of classical spinchains15. There, at any given time, either the even spinsin the chain were allowed to evolve under the influenceof the odd spins or vice versa. This provided access tolate times that would have been too difficult to access bydirect integration of the standard classical equations ofmotion.

By analogy, we propose allowing hopping on each bondin turn. At any given time, the instantaneous Hamilto-

4

nian looks like:

Ĥm = LamJ(ĉ†mĉm+1 + ĉ

†m+1ĉm)+

L−1∑j=0

[hj n̂j + V n̂j n̂j+1]

(3)We will specify the value of am in Section II.C below,where we discuss our choice of boundary conditions. Thestate of the system is allowed to evolve under this Hamil-tonian for a time ∆tL , and this evolution can be imple-mented by applying the unitary operator:

Ûm = exp

(−i∆t

LĤm

)(4)

One full time-step of duration ∆t consists of cyclingthrough all the bonds:

Û(∆t) =

L−1∏m=0

Ûm (5)

Note that, in (3), the hopping is enhanced by L becausethe hopping on any given bond is activated only once percycle, while the potential and interaction terms alwaysact. Therefore, the factor of L ensures that the aver-age Hamiltonian over a time ∆t has the form (1). Theadvantage of employing the modified dynamics is thatthe Ĥm only couple pairs of configurations, so prepar-ing the Ûm reduces to exponentiating order VH two-by-two matrices, where VH is the size of the Hilbert space.This is generally a simpler task than exponentiating theoriginal Hamiltonian (1). Our scheme only constitutes apolynomial speedup over exact diagonalization, but thatspeedup can increase the range of accessible lattice sizesby a few sites.

The modified dynamics raise several important is-sues that should be discussed51. The periodic time-dependence of the Hamiltonian induces so-called “multi-photon” (or “energy umklapp”) transitions betweenstates of the “parent” model (1) that differ in energy byωH =

2π∆t , reducing energy conservation to quasienergy

conservation modulo ωH . We need to question whetherthis destroys the physics of interest: does the single-particle Aubry-André transition survive, or do the multi-photon processes destroy the localized phase?

We take up this question in Appendix A, where wepresent a Floquet analysis of the single-particle andmany-body problems. We find that, for sufficientlysmall ∆t, the universal behavior of the single-particle AAmodel is preserved. At larger ∆t, multi-photon processescan strongly mix eigenstates of the single-particle parentmodel, increasing the single-particle density-of-states anddestroying the AA transition. In the spirit of the earlierreferenced work on classical spin chains15, our perspec-tive in this paper is to identify whether MBL can occurin a model qualitatively similar to our parent model (1).Therefore, to explore dynamics on long time scales, weavoid destroying the single-particle transition, but stillchoose ∆t to be quite large within that constraint.

In Appendix A, we also examine the consequencesof our choice of ∆t for the quasienergy spectrum ofthe many-body model. Our results suggest that multi-photon processes do not, in fact, strongly modify the par-ent model’s spectrum for much of the parameter rangethat we explore in this paper59. This means that partialenergy conservation persists in our simulations despitethe introduction of a time-dependent Hamiltonian, andwe need to keep this fact in mind when we analyze ournumerical data below.

Finally, we note in passing that several recent stud-ies have focused on the localization properties of time-dependent models52–54, including one on the quasiperi-odic Harper model55, but that the intricate details of thistopic are somewhat peripheral to our main focus.

C. Details of the Numerical Calculations

In studies of the 1D AA model, it is conventional toapproach the thermodynamic limit by choosing latticesizes according to the Fibonacci series (L = . . . 5, 8, 13,21, 34 . . .) and wavenumbers for the potential (2) as ra-tios of successive terms in the series26. These values of krespect periodic boundary conditions while converging tothe inverse of the golden ratio 1φ = 0.618033 . . .. For any

finite lattice, the potential is only commensurate with theentire lattice (since successive terms in the Fibonacci se-ries are mutually prime), and the duality mapping of theAA model is always exactly preserved. For our purposeshowever, this approach offers too few accessible systemsizes and complicates matters by generating odd valuesof L.

Instead, we found empirically that finite-size effects areleast problematic if we use exclusively even L, alwayskeep the wavenumber of the potential fixed at k = 1φ ,

and set:

am = 1− δm,L−1 (6)

in equation (3), thereby forbidding hopping over theboundary60. Note that, with these boundary conditions,our model describes hardcore bosons as well as fermions.The bosonic language maintains closer contact with coldatom experiments38; the fermionic language is more inkeeping with the MBL literature5,11.

Using the approach described above, we have simu-lated systems up to size L = 20 at half-filling. Oursimulations always begin with a randomly chosen con-figuration (or Fock) state, so that the initial state has noentanglement across any spatial bond in the lattice (i.e.each site is occupied or empty with probability 1). Ex-cept in the exact diagonalization studies of Appendix A,we always set ∆t = 1. We integrate out to tf = 9999 andultimately average the evolution of measurable quantitiesover several samples, where a sample is specified by thechoice of the initial configuration and offset phase to thepotential (2). The sample counts used in the numericsare provided in Table I.

5

L N VH samples

8 4 70 500

10 5 252 500

12 6 924 500

14 7 3432 250

16 8 12870 250

18 9 48620 250

20 10 184756 50

TABLE I: For the various simulated lattice sizes L, the parti-cle number N , the configuration space size VH , and the num-ber of samples used in the numerics. Note that we alwayswork at half-filling.

III. NUMERICAL MEASUREMENTS

We now introduce the quantities that we measure tocharacterize the different regimes of our model. Wealso present the numerical data along with some qual-itative remarks about the observed behavior. However,we largely defer quantitative phenomenology and model-ing of the data to Section IV.

A. Temporal Autocorrelation Function

One signature of localization is the system’s retentionof memory of its initial state. Since we simulate the re-versible evolution of a closed system, the quantum stateof the entire system retains full memory of its past. Nev-ertheless, we may still ask if the information needed todeduce the initial state is preserved locally or if it prop-agates to distant parts of the system. A diagnostic mea-sure with which to pose this “local memory” question isthe temporal autocorrelator of site j:

χj(t) ≡ (2〈n̂j〉(t)− 1)(2〈n̂j〉(0)− 1) (7)

Here, the angular brackets refer to an expectation valuein the quantum state. This single-site autocorrelator maybe averaged over sites and then over samples (as definedin Section II.C) to obtain:

χ(t;L) ≡

1L

L−1∑j=0

χj(t)

(8)The sample average is indicated here with the largesquare brackets. Typically, to reduce the effects of noise,we also average over a few time steps within each sam-ple (i.e. perform time binning) before taking the sampleaverage.

We can discriminate three qualitatively different be-haviors of χ vs. t in our interacting model. Figure 2shows examples of each of these behaviors at interac-tion strength u = 0.32. Panel (a) is characteristic ofthe low g regime, where the autocorrelator stays invari-ant over several orders-of-magnitude of time, and there is

no statistically significant difference between time seriesfor different L. At higher g, as in panel (b), the timeseries show approximately power-law decay culminatingin saturation to a late-time asymptote. For the largestsystems, the power law is roughly consistent with the dif-fusive expectation of t−

12 decay. The late-time asymptote

decays with L (as expected from energy conservation61 )suggesting that the power-law decay may continue indef-initely in the thermodynamic limit. Surprisingly, at stilllarger g, there is a third behavior, exemplified by panel(c). For the largest lattice sizes, the power-law era is notfollowed by saturation but by an extremely rapid decay.The rapid decay is most evident in the large g, large uregime, where the energy density of the parent model (1)is relatively large. This implies that this behavior mightbe tied to the multi-photon processes induced by peri-odic modulation of the Hamiltonian; correspondingly, italso implies that, for fixed g and u, we might be able toinduce the appearance of the rapid decay by increasing∆t. We have tested this numerically, and the results sup-port the connection to the energy non-conserving multi-photon processes. This suggests that there are only twodistinct regimes of the parent model represented in Fig-ure 2, differentiated by the L dependence of the asymp-totic value of the autocorrelator. We will proceed underthis working assumption.

The difference between these two regimes is broughtout more clearly in Figure 3. We focus on a late time t =ttest and probe χ(ttest;L) as a function of g for differentlattice sizes. Panels (a)-(c) show data for u = 0, 0.04,and 0.64 respectively. All the panels show a “splaying”point of the χ vs. L curves, separating a high g regimein which χ(ttest;L) decays with L from a low g regime inwhich it does not. The value of g at this feature decreasesmonotonically with u. Most importantly, in each case,this value is robust to changing ttest; if we halve ttest fromthe value that appears in Figure 3, the feature appearsat approximately the same value of g. This propertyof the data is very fortunate: in Section IV.C below,we will use the splaying feature in these plots to put anumerical lower bound on the transition value of g fordifferent interaction strengths. Since time scales get verylong near the transition, it is difficult to simulate out toconvergence in this regime. Nevertheless, the fact thatthe value of g at the splaying feature remains fixed in timeimplies that we can deduce the phase structure from ourfinite-time observations.

B. Normalized participation ratio

One of the commonly used diagnostics for studyingsingle-particle localization is the inverse participation ra-tio (IPR). This quantity is intended to probe whetherquantum states explore the entire volume of the systemand is often defined as the sum over sites of the amplitudeof the state to the fourth power:

∑j |ψj |4. Typically, the

IPR is inversely proportional to the localization volume

6

(a)

4 6 8 10−0.16

−0.14

−0.12

−0.1

−0.08

ln(t)

ln(

)

u = 0.32, g = 0.1

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

g

S 2/L

u = 0.04, t = 9999

8101214161820

(b)

4 6 8 10−5

−4

−3

−2

−1

ln(t)

ln(

)

u = 0.32, g = 0.5

(c)

4 6 8 10−10

−8

−6

−4

−2

ln(t)

ln(

)

u = 0.32, g = 0.85

FIG. 2: Three characteristic time series for the temporal auto-correlator with u = 0.32 and ∆t = 1. In each panel, we showtime series for a particular value of the hopping g. Only afew representative error bars are displayed in each time series.The legend refers to different lattice sizes L. The reference

lines in panels (b) and (c) show diffusive t−12 decay.

ξd in a single-particle localized phase and decays to zeroas the inverse of the system volume in an extended phase.

We now describe how this quantity can be fruitfullyexploited in the many-body context. Let c denote somespecific configuration of N particles in L sites. Then,we can write the state of the system in the configurationbasis as:

|Ψ(t)〉 =∑{c}

ψc(t) |c〉 (9)

The configuration-basis IPR is simply:

P (t;L) ≡

[∑c

|ψc(t)|4]

(10)

where the square brackets, as usual, denote a sample av-erage. Interpreting P (t;L) as the inverse of the numberof configurations on which |Ψ(t)〉 has support, we nowdefine the normalized participation ratio (NPR):

η(t;L) ≡ 1P (t;L)VH

(11)

(a)

0 0.5 10

0.5

1

g

u = 0, tbin = 9980−9999

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

g

S 2/L

u = 0.04, t = 9999

8101214161820

(b)

0 0.5 10

0.5

1

g

u = 0.04, tbin = 9980−9999

(c)

0 0.5 10

0.5

1

g

u = 0.64, tbin = 9980−9999

FIG. 3: The value of χ in the latest time bin (t =9980 . . . 9999) plotted against g. In panels (a)-(c), u = 0,0.04, and 0.64 respectively. The legend refers to different lat-tice sizes L.

The quantity η(t;L) then represents the fraction of con-figuration space that the system explores. We expectη(t;L) to be independent of L at late times in the er-godic phase. In the many-body localized phase, we ex-pect η(t;L) to decay exponentially with L.

In Figure 4, we plot η(ttest;L) vs. g for u = 0, 0.04, and0.64. The figure reveals an important difference betweenthe non-interacting and interacting models. At low g,both with and without interactions, η decays exponen-tially with L:

η ∝ exp(−κL) (12)

with κ > 0. More surprisingly, η also decays with L atlarge g in the non-interacting case; all that happens isthat κ becomes essentially independent of g. With evensmall interactions however, η becomes system-size inde-pendent in the large g regime, following our ansatz for anergodic phase. We bring out this point more clearly inFigure 5, in which we extract estimates for the decay co-efficient κ for various values of the interaction strength.Thus, the extended phase of the non-interacting AAmodel appears to be a special, non-ergodic limit.

7

(a)

0 0.5 1−15

−10

−5

0

g

ln(

)

u = 0, tbin = 9980−9999

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

g

S 2/L

u = 0.04, t = 9999

8101214161820

(b)

0 0.5 1−15

−10

−5

0

g

ln(

)

u = 0.04, tbin = 9980−9999

(c)

0 0.5 1−15

−10

−5

0

g

ln(

)

u = 0.64, tbin = 9980−9999

FIG. 4: The value of η in the latest time bin (t =9980 . . . 9999) plotted against g. In panels (a)-(c), u = 0,0.04, and 0.64 respectively. The legend refers to different lat-tice sizes L. See equation (11) for the definition of η. In theergodic phase η ≈ 0.5.

0.2 0.4 0.6 0.8−0.5

0

0.5

g

tbin = 9980−9999

0 200 400 600 800 1000

0.35

0.4

0.45

0.5

0.55

g

00.040.160.320.64

FIG. 5: Estimates of κ from a fit of η ∝ e−κL in the latesttime bin (tbin = 9980 − 9999). The legend refers to differentvalues of the interaction strength u.

Before proceeding, we should caution that, in panels(b) and (c) of Figure 4, the collapse at high g looks veryappealing because of the use of a semilog plot and wouldnot be so striking on a normal scale. The axes havebeen chosen to highlight the exponential scaling at lowg, which would not be as apparent if we simply plottedη vs. g. However, regarding the absence of perfect col-lapse at high g, note that the raw data for the IPR differby several orders-of-magnitude for different values of thelattice size L. Given this, the coincidence of the order-of-magnitude of η for different values of L is already a goodindication of the proposed scaling, and some correctionsto this scaling should be expected given the modest ac-cessible system sizes.

C. Rényi Entanglement Entropy

Unlike the normalized participation ratio, which pro-vides a global characterization of the time-evolved state,bipartite entanglement is arguably a better proxy forwhether a part of the system can act as a good heatbath for the rest. In the many-body ergodic phase, weexpect the bipartite entanglement entropy to be a faith-ful reflection of the thermodynamic entropy. This im-plies an extensive entropy, pinned to its thermal infinitetemperature value throughout the phase62. In contrast,in the many-body localized phase, we expect an exten-sive but subthermal entanglement entropy. This expec-tation is consistent with the results of three recent papersthat focus on the behavior of entanglement measures inthe many-body localized phase of the disordered prob-lem13,19,20 . These papers also study the time dependenceof the entropy beginning from an unentangled productstate. In the many-body localized phase, this growth isfound to be slow, generically logarithmic in time. Sinceour model lacks disorder altogether, it may be interest-ing to explore the entanglement dynamics here as well.In what follows, we comment on the dynamics, but weprimarily use the late-time entanglement entropy as yetanother tool to help distinguish between the many-bodylocalized and ergodic phases.

Let subsystem A refer to lattice sites 0, 1, . . . L2 − 1,and let subsystem B refer to the remaining sites in thechain. We can compute the reduced density matrix ofsubsystem A by beginning with the full density matrixρ̂(t) = |Ψ(t)〉 〈Ψ(t)| and tracing out the degrees of free-dom associated with subsystem B:

ρ̂A(t) ≡ TrB{ρ̂(t)} (13)

The sample-averaged order-2 Rényi entropy of subsystemA is then given by:

S2(t;L) ≡[− log2

(TrA{ρ̂A(t)2}

)](14)

Both S2 and the standard von Neumann entropy are ex-pected to attain the same values in the ergodic phase;

8

(a)

0 2 4 6 8 100

0.2

0.4

0.6

ln(t)

S 2

g = 0.2

0 2000 4000 6000 8000 100000

0.5

1

1.5

ln(t)

S 2

g = 0.2

00.160.64

(b)

0 2 4 6 8 100

5

10

ln(t)

S 2

g = 1.1

FIG. 6: Example time series of the Rényi entropy for two val-ues of the tuning parameter g. The legend refers to differentvalues of the interaction strength u. Panel (a) shows data forL = 10 lattices at g = 0.2. Panel (b) shows data for L = 20lattices at g = 1.1. In the localized regime, we need to usesmaller lattices to see convergence Renyi entropy.

we choose to focus on the former to save on the compu-tational cost of diagonalizing the reduced density matrix(13).

Our first task is to examine whether the putative local-ized phase of our model exhibits the same behavior thatwas observed with tDMRG13,19. In panel (a) of Figure6, we focus on a low value of g and plot S2 vs. ln(t)for L = 10 lattices. At very early times, the time seriesall tend to coincide, reflecting the formation of short-range entanglement at the cut between the subsystems.Afterwards, the non-interacting time series saturates forseveral orders-of-magnitude of time, while the interactingtime series show behavior that is consistent with logarith-mic growth. In order to clearly establish the saturationthat follows the slow growth, we have had to focus onsmall lattices. Panel (b) of Figure 6 shows data for largeg. Here, the most striking difference between the non-interacting and interacting models lies in the saturationvalue of the entropy: the interacting model is substan-tially more entangled, but the saturation value does notappear to depend on the value of u. We will see belowthat this is another indication that thermalization onlyoccurs in the interacting, large g regime.

Figure 7 shows late-time values of the Rényi entropydensity plotted against the tuning parameter g. We firstfocus on the high g regime. In panel (a), u = 0, andS2(ttest;L) ∝ L for large g. However, the entropy den-sity is less than 12 , which is the thermal result when thesystem has ergodic access to all configurations consistentwith particle number conservation. The situation is dra-

matically different in panels (b) and (c), where u = 0.04and 0.64 respectively. At high g, the entropy actuallylooks superextensive. This is just a finite-size effect, be-cause the entropy is well fit to a linear growth of theform:

S2(ttest;L) = mL− Sdef (15)

where Sdef is a constant deficit, typically around 1.15 −1.3. In Figure 8, we show that the slope m ≈ 12 at large gin the interacting problem. This implies that the entropyis thermal in the L → ∞ limit, where the deficit Sdef isnegligible.

Now, we turn to the low g regime. Without inter-actions, the off-diagonal elements in the reduced densitymatrix (13) typically contain only a few frequencies origi-nating from localized single particle orbitals immediatelyadjacent to the cut. The number of relevant orbitals isfinite in L. As a result, the off-diagonal elements can-not fully vanish, and the reduced density matrix neverthermalizes. The resulting entanglement entropy is inde-pendent of L as shown in the inset of panel (a). In theinteracting problem, while the orbitals immediately ad-jacent to the cut still have roughly the same frequencies,the “spectral drift” (i.e. the spread of these lines dueto sensitivity to the configuration of distant particles)allows for a much larger number of distinct and mutu-ally incoherent contributions to offdiagonal elements ofthe reduced density matrix. These off-diagonal elementscan dephase more efficiently, leading to a partial ther-malization. This is the mechanism that likely underliesthe extensive but subthermal entropy observed by Bar-darson et al.19. For small L, our numerical results in thelow g regime agree well with this expectation. For largerL, the slow dynamics of the entropy formation makes itdifficult to observe saturation, both in our work and inthe tDMRG study of Bardarson et al.

If the entropy eventually becomes extensive for all L,then the “crossing” feature that is present in panels (b)and (c) of Figure 7 would become a “splaying” feature,with the entropy density independent of L at small g. Inany case, an interesting property of the data is that thevalues of g at the crossing features of the S2(ttest;L) vs.g plots are consistent with the locations of the splayingfeatures in the corresponding χ(ttest;L) vs. g plots ofFigure 3. This seems to be the case for all u. Thus, thesefeatures may be useful in locating the transition.

IV. MODELING THE MANY-BODY ERGODICAND LOCALIZED PHASES

Above, we presented numerical evidence that our in-teracting AA model contains two regimes that show qual-itatively distinct behavior of the autocorrelator, normal-ized participation ratio, and Rényi entropy. Next, wewill propose and characterize model quantum states thatqualitatively (and sometimes quantitatively) reproducethe numerically observed late-time behavior in the two

9

(a)

0 0.5 10

0.1

0.2

0.3

g

S 2/L

u = 0, t = 9999

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

g

S 2/L

u = 0.04, t = 9999

8101214161820 0 0.1 0.2 0.3 0.40

1

2

g

S 2

u = 0, t = 9999

(b)

0 0.5 10

0.2

0.4

g

S 2/L

u = 0.04, t = 9999

0 0.2 0.40

0.1

0.2

g

S 2/L

u = 0.04, t = 9999

(c)

0 0.5 10

0.2

0.4

g

S 2/L

u = 0.64, t = 9999

0 0.2 0.40

0.2

0.4

g

S 2/L

u = 0.64, t = 9999

FIG. 7: The value of S2L

at t = 9999 plotted against g. Inpanels (a)-(c), u = 0, 0.04, and 0.64 respectively. The legendrefers to different lattice sizes L. In panel (a), the inset plotshows S2 vs. g in the low g regime. In panels (b) and (c), theinsets show S2

Lvs. g for low L in the low g regime.

0.2 0.4 0.6 0.8−0.20

0.20.40.6

g

m

t = 9999

m = 1/2

m = 00 200 400 600 800 1000

0.35

0.4

0.45

0.5

0.55

g

00.040.160.320.64

FIG. 8: The estimated slope of S2 vs. L at late times asa function of g. The legend refers to different values of theinteraction strength u.

regimes. These model states expose more clearly whythe two regimes of our model are appropriately identifiedas many-body ergodic and localized phases.

A. The Many-Body Ergodic Phase

To model the behavior of the putative ergodic phase,we begin by writing down a generic model state in theconfiguration basis:

|Φ〉 =∑{c}

φc |c〉 =L2∑

n=0

∑{cA,cB}

φ(n)AB

∣∣∣c(n)A , c(n)B 〉 (16)Here, the c refer to configurations of the full chain,whereas the cA and cB refer to configurations of the sub-systems A and B, as defined in Section III.C above. Thesuperscripts on the configurations and expansion coeffi-cients refer to the number of particles in subsystem A.Writing the state in terms of the subsystem configura-tions will be useful shortly, but for now we focus on thestatistical properties of the amplitude φc. We assumethat this amplitude is distributed as a complex Gaussianrandom variable:

p(φ) =1

2πσ2exp

(−|φ|

2

2σ2

)(17)

Within this distribution, 〈|φ|2〉 = 2σ2 and 〈|φ|4〉 = 8σ4.From these average values, it is possible to deduce that:

σ =1√2VH

(18)

for normalization and that the IPR is PΦ =2VH

. This, inturn, implies:

ηΦ =1

2(19)

This result is reproduced quantitatively in the numericsin Figure 4.

Next, suppose we compute the reduced density matrixof subsystem A in the state |Φ〉:

ρ̂A =∑n

∑{cA,cA′ ,cB}

φ∗(n)AB φ

(n)A′B

∣∣∣c(n)A 〉〈c(n)A′ ∣∣∣ (20)To find the Rényi entropy, we need to compute the traceof the square of this operator:

TrA{ρ̂2A} =∑n

∑{cA,cA′ ,cB ,cB′}

φ∗(n)AB φ

(n)A′Bφ

∗(n)AB′φ

(n)A′B′

(21)When we average over our distribution of amplitudes(17), only the coherent terms survive63:

TrA{ρ̂2A} ≈∑n

∑{cA,cB ,cB′}

〈|φ(n)AB |2|φ(n)AB′ |

2〉

10

+∑n

∑{cA,cA′ ,cB}

〈|φ(n)AB |2|φ(n)A′B |

2〉

−∑n

∑{cA,cB}

〈|φ(n)AB |4〉 (22)

The final term accounts for the double counting of termswhere cA = cA′ and cB = cB′ simultaneously. We nowintroduce the notation:

γ(P,Q) =P !

Q!(P −Q)!(23)

and evaluate the expectation values in equation (21) toobtain:

TrA{ρ̂2A} ≈2

V 2H

∑n

γ

(L

2, n

)3(24)

Finally, using a Stirling approximation to the combina-tion function and a saddle-point approximation for thesum, we find the entropy:

S2,Φ ≈L

2− log2

(4√3

)≈ L

2− 1.2 (25)

This is the same form observed in the numerics (15), andthe deficit Sdef lies in the observed range. Asymptoticallyin L, the entropy (25) is maximal, and this is exactly theexpected behavior when the particle number thermalizes.

There is an important caveat to note here: we have ar-gued above that, if multi-photon processes do not com-pletely destroy energy conservation, then this can leadto relic autocorrelations at late times. This implies thatthe assumption of independent random amplitudes can-not be exactly correct on a finite lattice. However, thenumerically-observed relic autocorrelations decay with L,suggesting that our assumptions get better as the systemsize grows. Therefore, in the thermodynamic limit, thisphase is truly thermal.

B. The Many-Body Localized Phase

Our model for the time-evolved state in the localizedregime is founded upon the following intuition: there ex-ists a length scale ξ, which is analogous to the single-particle localization length and beyond which particlesare unlikely to stray from their positions in the initialstate. Then, if we partition our lattice of length L intoblocks of size ξ, exchange of particles between blocksis less important than rearrangements of the particleswithin each block. Consequently, the total number ofconfigurations accessed by the state of the full systemis approximately the product of the number of configu-rations accessed within each block. This multiplicativeassumption should be very safe in a localized phase. Weadditionally assume that, within each block, the dynam-ics completely scramble the particle configuration. If acertain block of length ξ contains n particles in the initial

state, then the time-evolved state contains equal ampli-tude for each of the possible ways of arranging n particlesin those ξ sites. In keeping with our numerical protocol,we randomly select the initial state from the space of allpossible Fock states of a certain global particle number.Then, a block of ξ sites contains n particles with proba-bility:

w(ξ, n) =γ(ξ, n)

2ξ

[1 +O

(ξ2

L

)](26)

We will consider the limit L � ξ � 1, where we canapproximate the probability by the first term. The as-sumptions proposed above motivate writing down a stateof the form:

|Λ〉 = 1√M

∼∑{c1,...cL

ξ}

z(c1, . . . cL

ξ

) ∣∣∣c1, . . . cLξ

〉(27)

where the tilde on the sum indicates that it should onlyrun over configurations that are consistent with the initialdistribution of particles among the blocks. The factors zare complex phases which depend upon the configuration,and M is a normalization which is equal to the totalnumber of configurations represented in the state |Λ〉.

Before beginning our analysis of the state |Λ〉, weshould note that, in contrast to our calculations in theergodic phase, our goal in the localized regime will beto qualitatively tie the numerically observed large L be-havior to the existence of the length scale ξ. Unfortu-nately, we cannot achieve the quantitative accuracy ofthe ergodic model state |Φ〉 with the localized toy-modeldescribed above.

We begin by estimating the autocorrelator between theinitial state and the model time-evolved state |Λ〉. A non-zero autocorrelator emerges, because each block is onlyat half-filling on average. Fluctuations away from half-filling (in either direction) yield a positive typical value ofthe autocorrelator within a block. Indicating an averageover the distribution (26) with an overline, we find theblock value χblock ≈ 1L . This is also the average value forthe whole system when L� ξ:

χΛ ≈1

ξ(28)

Next, to estimate the IPR, we need to compute the nor-malization factor M . We begin by estimating the numberof explored configurations in each block. The average ofthe logarithm of the number of explored configurationswithin a block is:

ln(Mblock) ≈ ln(√

2

πξ2ξ)− 1

2(29)

Then, using lnM ≈ Lξ lnMblock, we can estimate M itselfas:

M ≈ elnM ≈ 2L(πeξ

2

)− L2ξ(30)

11

Using this normalization, we can estimate the NPR ηΛ:

ln ηΛ ≈ −L

2ξln

(πeξ

2

)+

1

2lnL+

1

2ln(π

2

)(31)

This qualitatively agrees with the numerically observedbehavior (12) up to subleading corrections, and in thelarge-L limit:

κ ≈ 12ξ

ln

(πeξ

2

)(32)

Note that equations (28) and (32) imply a relationshipbetween the scaling behaviors of χ and κ in the localizedregime. This relationship is not reflected in our numericaldata, in part because we cannot truly attain the limitL � ξ � 1. The numerically computed value of κ, forexample, can contain finite-size corrections of order ln(L)Lor ξ

2

L . Also, we must keep in mind that the state |Λ〉is just a toy model that does not capture fine details ofthe time-evolved states in this regime. Thus, we must becontent with reproducing the qualitative behavior of eachmeasurable quantity individually, without expecting therelationships between these quantities in |Λ〉 to be exactlyreproduced in the data.

We now turn to the Rényi entropy, the quantity whichmost strikingly distinguishes between the non-interactingand interacting localized phases. To examine this quan-tity, we revert to partitioning the system in half, insteadof into blocks of size ξ. As long as ξ � L2 , the assump-tions that we made above about the blocks of size ξ holdeven better for the subsystems A and B. For example, wecan assume that there are “explored sets” of MA config-urations in subsystem A and MB configurations in sub-system B respectively, with M = MAMB . We considercomputing the reduced density matrix ρ̂A, exactly as inequation (20) above. If the off-diagonal elements of thisdensity matrix remain perfectly phase-coherent, it caneasily be shown that Scoh2,Λ = 0. In reality, there will be alocal contribution to the entropy from particles strayingover the cut between subsystems A and B. This mim-ics the situation in non-interacting localized phases. Al-ternatively, suppose that dephasing is sufficiently strongthat we can proceed by analogy with the ergodic phase,beginning with equation (21) and keeping only coherentterms as in equation (22). Thereafter, the calculation forthe model localized state |Λ〉 differs from the calculationfor |Φ〉. We need to consider the statistics of the con-figuration probabilities |λAB |2. For |λAB |2 6= 0, we needthe configurations on both subsystems to lie within theirrespective explored sets; this occurs in subsystem A, forexample, with probability MA

γ(L2 ,n). This reasoning leads

to the “dephased” entropy:

Sdp2,Λ ≈ − log2(

1

MA+

1

MB− 1MAMB

)≈ − log2

(2√M− 1M

)

≈ 12

[1− 1

2ξlog2

(πeξ

2

)]L− 1 (33)

where we have additionally made the approximation thattypically MA ≈MB ≈

√M . With only partial loss of co-

herence, the entropy would lie between these two limiting

cases: Scoh2,Λ ≤ S2,Λ ≤ Sdp2,Λ. Thus, dephasing alone, with-

out additional particle transport, can induce an extensiveentropy.

Indeed, our numerics support the view that the maindifference between the non-interacting and many-bodylocalized phases is the amount of dephasing. There doesnot seem to be a qualitative difference in particle trans-port. The particle configuration stays trapped near itsinitial state, even with interactions, and the system doesnot thermalize.

V. TRACING THE PHASE BOUNDARY

in this section, we use the data from Section III toextract estimates of the phase boundary between the lo-calized and ergodic phases. Estimating the location ofthe MBL transition is extremely challenging. Given thenumerically accessible lattice sizes, satisfying finite-sizescaling analyses are difficult to perform. Nevertheless,rough estimates have been made in the disordered prob-lem11,12,16,21, so we will now attempt to extract an ap-proximate phase boundary for our model.

We first consider the autocorrelator. Above, we notedthe “splaying” feature in the late-time plots of the auto-correlator vs. g. The value of g at this feature can betaken as a lower bound for the transition. For g slightlygreater than this value, it is possible that χ only decayswith L because ξ > L for accessible lattice sizes. Consid-ering two lattice sizes (L = 16 and 20) and finding whentheir values of χ deviate, we find the values reported inthe first column of Table II.

Next, we consider the fitting parameter κ in equation(12). In Figure 5, we see that there is a region whereκ < 0 for finite interaction strength. Since η ≤ 1, finite-size effects are clearly dominating the estimate in thisregion. We can use the value of g where κ is minimal totrack how this region moves as u is varied. This yieldsthe second column of the table.

Finally, a similar approach can be applied to extractestimates of gc from the fits (15). There exists a regionwhere m > 12 , but this is mathematically inconsistent inthe thermodynamic limit. Therefore, if we find the valueof g that maximizesm, we can again estimate the locationof the region dominated by finite-size effects, yielding thefinal column of Table II.

The estimates of the transition value gc in Table IIwere obtained using data for the latest time that we sim-ulated (the time bin tbin = 9980 . . . 9999 for χ and κ andt = 9999 for m). However, we have also estimated gc fordata obtained at a half and a quarter of this integrationtime, finding consistent results. Thus, the general phase

12

u χ κ m

0.04 0.35 0.45 0.45

0.16 0.30 0.40 0.40

0.32 0.25 0.40 0.40

0.64 0.25 0.40 0.35

TABLE II: Bounds or estimates of the transition value of gc atvarious values of u and based on various measured quantities.The column titled χ gives a lower bound on the transitionvalue of g based on the autocorrellator. The remaining twocolumns give estimates of gc based on κ and m, as defined inSections III.B and III.C respectively. See Section V for thereasoning behind the estimates. All values carry implicit errorbars of ±0.05 as that is the discretization of our simulatedvalues of g. This error bar should be interpreted, for instance,as the error on our estimate of the location of the maximumvalue of m. The error on our estimate of gc is, of course, muchlarger.

structure of the model is invariant to changing the obser-vation time, even though not all measurable quantitieshave converged to their asymptotic values. Consolidat-ing the information from the estimates in Table II, wepropose that the phase diagram qualitatively resemblesFigure 1.

Before proceeding, it is worth noting that our roughestimates of the phase boundary do not make assump-tions regarding the character of the MBL transition (i.e.whether it is continuous or first order). In fact, some ofour plots (e.g. panel (c) of Figure 7) hint at the possi-bility of a discontinuous change in S2 as a function of gin the thermodynamic limit. We are not aware of anyresults that rule out a first-order MBL transition, so wemust keep this possibility in mind.

VI. CONCLUSION

Recently, evidence has accumulated that Ander-son localization can survive the introduction of suffi-ciently weak interparticle interactions, giving rise toa many-body localization transition in disordered sys-tems5,6,11,12,21. The MBL transition appears to be athermalization transition: in the proposed many-bodylocalized phase, the fundamental assumption of statisti-cal mechanics breaks down, and the system fails to serveas its own heat bath11,12. We have presented numeri-cal evidence that this type of transition can also occurin systems lacking true disorder if they instead exhibit“pseudodisorder” in the form of a quasiperiodic poten-tial.

From one perspective, this may be an unsurprisingclaim. For g < 12 the localized single-particle eigenstatesof the quasiperiodic Aubry-André model have the samequalitative structure as those of the Anderson model, sothe effects of introducing interactions ought to be similar.By this reasoning, perhaps it is even possible to guess thephase structure of an interacting AA model using knowl-

edge of an interacting Anderson model: we simply matchlines of the two phase diagrams that correspond to thesame non-interacting, single-particle localization length.

However, this perspective misses important effects inall regions of the phase diagram. Most obviously, the AAmodel has a transition at u = 0, and it is interesting tosee how this transition gets modified as it presumablyevolves into the MBL transition at finite u. It is alsoimportant to remember that quasiperiodic potentials arecompletely spatially correlated. This means that the AAmodel lacks rare-regions (Griffiths) effects, and this mayhave subtle consequences for the dynamics. Finally, theAA model contains a phase that is absent in the one-dimensional Anderson model, the g > 12 extended phase,and we have seen above that interactions have a profoundeffect upon this regime.

Understanding MBL in the quasiperiodic context is es-pecially pertinent given the current experimental situa-tion. Some experiments that probe localization physicsin cold atom systems use quasiperiodic potentials, con-structed from the superposition of incommensurate opti-cal lattices, in place of genuine disorder. The group of In-guscio, in particular, has recently explored particle trans-port for interacting bosons within this setup38,39. Mean-while, the AA model has also been realized in photonicwaveguides, and experimentalists have studied the effectsof weak interactions on light propagation through thesesystems. They have also investigated “quantum walks”of two interacting photons in disordered waveguides40,56.This protocol resembles the one we have implemented nu-merically, so similar physics may arise. Finally, we notethat Basko et al. have predicted experimental manifes-tations of MBL in solid-state materials. In such systems,there is always coupling to a phononic bath, so the MBLtransition is expected to become a crossover that nev-ertheless retains interesting manifestations of the MBLphenomena57. Whether there exist quasiperiodic solid-state systems to which the predictions of Basko et al.apply remains to be understood.

Given the current experimental relevance of localiza-tion phenomena in quasiperiodic systems, we hope thatour study will motivate further attempts to understandthese issues. Unfortunately, our ability to definitivelyidentify and analyze the MBL transition is limited bythe modest lattice sizes and evolution times that we cansimulate. Vosk and Altman recently developed a strong-disorder renormalization group for dynamics in the disor-dered problem20, but the reliability of such an approachin the quasiperiodic context is unclear. A time-dependentdensity matrix renormalization (tDMRG) group studyof this problem would be a valuable next step. Tezukaand Garćıa-Garćıa have published tDMRG results on lo-calization in an interacting AA model, but their focuswas not on the thermalization questions of many-bodylocalization44. It would be worthwhile to pose these ques-tions using a methodology that allows access to muchlarger lattices. However, even tDMRG may have diffi-culty capturing the highly-entangled ergodic phase13,19,

13

so an effective numerical approach for definitively char-acterizing the transition remains elusive.

Acknowledgments

We thank E. Altman, M. Babadi, E. Berg, S.-B.Chung, K. Damle, D. Fisher, M. Haque, Y. Lahini,A. Lazarides, M. Moeckel, J. Moore, A. Pal, S.Parameswaran, D. Pekker, S. Raghu, A. Rey and J. Si-mon for helpful discussions. This research was supported,in part, by a grant of computer time from the City Uni-versity of New York High Performance Computing Cen-ter under NSF Grants CNS-0855217 and CNS-0958379.S.I. thanks the organizers of the 2010 Boulder School forCondensed Matter and Materials Physics. S.I. and V.O.thank the organizers of the Cargesè School on DisorderedSystems. S.I. and G.R. acknowledge the hospitality of theFree University of Berlin. V.O. and D.A.H are gratefulto KITP (Santa Barbara), where this research was sup-ported in part by the National Science Foundation underGrant No. NSF PHY11-25915. V.O. thanks NSF forsupport through award DMR-0955714, and also CNRSand Institute Henri Poincaré (Paris, France) for hospi-tality. D.A.H. thanks NSF for support through awardDMR-0819860.

Appendix A: Exact Diagonalization Results for theSingle-Particle and Many-Body Problems

This appendix collects exact diagonalization resultsthat supplement the real-time dynamics study in themain body of the paper.

1. Floquet Analysis of the Modified Dynamics

The goal of the first part of this appendix is to exam-ine the consequences of the modifications to the quantumdynamics described in Section II.B above. We first ver-ify that the AA transition survives by diagonalizing thesingle-particle AA Hamiltonian (i.e. the Hamiltonian (1)with u = Vh = 0) and the single-particle unitary evolu-tion operators (5) for various choices of the time step ∆t.Subsequently, we employ the same approach to examinehow varying ∆t impacts the quasienergy spectrum of theinteracting, many-body model.

a. Robustness of the Single-Particle Aubry-AndréTransition

To study the single-particle transition, we focus on theinverse participation ratio:

Psp(g;L) =

L−1∑j=0

|ψj |4 (A1)

(a)

−200 −100 0 100 2000

10

20

(g−0.5)L

P sp/

L0.5

AA model

−200 −150 −100 −50 0 50 100 150 2000

0.5

1

1.5

2x 104

(g−0.5)L

P sp/

L0.5

AA model

8163264128256512

−50 0 500

5

10

(g−0.5)L

P sp/

L0.5

AA model

(b)

−100 0 100 2000

10

20

(g−0.45)L

P sp/

L0.5

t = 1

−50 0 500

5

10

(g−0.45)L

P sp/

L0.5

t = 1

FIG. 9: Collapse of single-particle IPR vs. g, using the scalinghypothesis (A2). The legend refers to different lattice sizes L.In panel (a), we show data for the usual AA Hamiltonian (1).In panel (b), we show data obtained from diagonalizing theunitary evolution operator for one time step in the modifieddynamics (5). We use potential wavenumber k = 1

φand 50

samples for all lattice sizes. The insets show magnified viewsof the curves for the three largest lattice sizes in the vicinityof the transition.

Here, ψj denotes the amplitude of the wave function atsite j of an L site lattice. We enclose the sum in equation(A1) in parentheses to indicate important differences inthe averaging procedure with respect to the many-bodyinverse participation ratio (10). In the many-body case,we computed the IPR as a sum over configurations in thequantum state at a particular time in the real-time evo-lution. Then, we averaged over samples, where a samplewas specified by a choice of the offset phase to the po-tential (2) and an initial configuration. Throughout thisappendix, we instead specify a “sample” solely by the off-set phase δ, and we average over eigenstates within eachsample before averaging over samples.

As noted previously, the usual AA model has a tran-sition that must occur, by duality, at gc =

12 . Near the

transition, the localization length is known to divergewith exponent ν = 126. Our exact diagonalization re-sults indicate that, at the transition, Psp(gc, L) ∼ L−

12 .

Hence, we can make the following scaling hypothesis forthe IPR:

Psp = L− 12 f((g − gc)L) (A2)

In panel (a) of Figure 9, we show that we can use thisscaling hypothesis to collapse data for the standard AAmodel. We show data for L = 8 to L = 512, with poten-tial wavenumber k = 1φ and open boundary conditions.

For all lattice sizes, we average over 50 samples.

14

(a)

−5 0 50

0.02

0.04

0.06

0.08

E

d(E)

L = 12, g = 0.25, u = 0.16

0 200 400 600 800 10000

0.02

0.04

0.06

0.08

E

P(E)

L = 8, g = 0.25, u = 0.16

0.1250.250.5124PM

(b)

−5 0 50

0.02

0.04

0.06

0.08

E

d(E)

L = 12, g = 0.4, u = 0.16

(c)

−5 0 50

0.02

0.04

0.06

0.08

E

d(E)

L = 12, g = 0.9, u = 0.16

FIG. 10: The density-of-states vs. quasienergy for L = 12systems at half-filling with interaction strength u = 0.16. Thelegend refers to different values of ∆t; the time-independent,parent model is referred to as “PM.” In panels (a)-(c), g =0.25, 0.4, and 0.9 respectively.

To establish the stability of the AA transition to themodified dynamics, we must ask: can the IPR obtainedfrom diagonalizing the unitary evolution operators (5) bedescribed using the scaling hypothesis (A2)? Panel (b)of Figure 9 shows that this is indeed the case for ∆t = 1.The only parameter that needs to be changed is gc, whichdecreases slightly as ∆t is raised. This implies that thereis a transition in the Floquet spectrum of the system thatcan be tuned by varying ∆t. It would be a worthwhileexercise to map out the phase diagram of this single-particle problem in the (g,∆t) plane. We leave this forfuture work.

b. Properties of the Many-Body Quasienergy Spectrum

We now turn our attention back to the effects of themodified dynamics upon the full, many-body model.In Section II.B above, we emphasized that our time-dependent model lacks energy conservation, with multi-photon processes inducing transitions between states ofthe parent model (1) that differ in energy by ωH =

2π∆t .

In this part of the appendix, we will examine how vary-ing ∆t impacts the quasienergy spectrum of the time-dependent model, using the approach that we applied tothe single-particle case above: we diagonalize the time-independent Hamiltonian as well as the unitary evolutionoperator for one time step of the time-dependent model.

In Figure 10, we plot the density-of-states d(∆t, E)in quasienergy space of the parent model and time-dependent models for different values of ∆t. We focuson L = 12 systems at half-filling with fermions (or, sincewe continue to use the boundary conditions describedin Section II.C, hardcore bosons). We fix the interac-tion strength to u = 0.16 and tune g to explore differentregimes of the model. In panels (a)-(c), we plot data forg = 0.25, 0.4, and 0.9. According to Table II, these val-ues of g put the system in the localized phase, near thetransition, and in the ergodic phase respectively.

We first consider the consequences of varying ∆t whileholding the other parameters fixed. For sufficiently small∆t, the quasienergy spectrum faithfully reproduces allthe structure of the energy spectrum of the parent model.This is unsurprising, because if ωH is greater than thebandwidth of the parent model’s spectrum, direct multi-photon processes will not take place. If we now tune ωHso that it is less than this bandwidth, the quasienergyspectrum begins to deviate from the parent model’s spec-trum at its edges. This effect can be seen, for instance, byexamining the trace for ∆t = 1 in panels (a) or (b). Foreven higher values of ∆t (i.e. lower values of ωH), multi-photon processes strongly mix the states of the parentmodel, resulting in a flat quasienergy spectrum.

The effect of multi-photon processes can also be en-hanced by broadening the parent model’s spectrum,which can be achieved by raising g or u. In panel (c) ofFigure 10 for instance, multi-photon processes have sig-nificantly flattened the spectrum for ∆t = 1, and devia-tions from the parent model are even visible for ∆t = 0.5.Since we always use ∆t = 1 in our real-time dynamicssimulations, it is perhaps fortunate that g = 0.9 is wellwithin the proposed ergodic phase for u = 0.16 and that,near the critical point (i.e. in panel (b)), the quasienergyspectrum for ∆t = 1 still retains much of the structureof the parent model’s spectrum.

However, there is one more caveat to keep in mind:the energy content of the system also grows with L. Atfixed g, u, and ∆t, the properties of the parent andtime-dependent models deviate from one another as thesystem size grows. If we truly want to faithfully repro-duce the dynamics of the parent model with the modifieddynamics, it may be necessary to scale ∆t down as weraise L. However, recall that our goal is simply to findMBL in a model qualitatively similar to the parent model(1). Even with this more modest goal in mind, there isstill the danger that, on sufficiently large lattices, multi-photon processes might couple a very large number oflocalized states and thereby destroy the many-body lo-calized phase of the parent model. Our numerical obser-vations indicate that this does not happen for the sys-

15

tem sizes that we can simulate. We can keep ∆t fixed atunity for L ≤ 20 without issues, accepting the possibilitythat the sequence of models that we would in principlesimulate on still larger lattices may require progressivelysmaller values of ∆t.

2. Level Statistics of the Many-Body Parent Model

Localization transitions are often characterized bytransitions in the level statistics of the energyspectrum50. Two of us previously looked at the levelstatistics of the disordered problem and identified acrossover from Poisson statistics in the many-body lo-calized phase to Wigner-Dyson statistics in the many-body ergodic phase11. The intuition that underlies thiscrossover is the following: in a localized phase, particleconfigurations that have similar potential energy are toofar apart in configuration space to be efficiently mixedby the kinetic energy term in the Hamiltonian. There-fore, level repulsion is strongly suppressed, and Poissonstatistics hold. Conversely, in an ergodic phase, there isstrong level repulsion which lifts degeneracies, leading toWigner-Dyson (i.e. random matrix) statistics.

Along the lines of the aforementioned study of the dis-ordered problem, we focus on the gaps between succes-sive eigenstates of the spectrum of the many-body parentmodel (1):

δn ≡ En+1 − En (A3)

and a dimensionless parameter that captures the corre-lations between successive gaps in the spectrum:

rn ≡min(δn, δn+1)

max(δn, δn+1)(A4)

For a Poisson spectrum, the rn are distributed as2

(1+r)2

with mean 2 ln(2) − 1 ≈ 0.386; meanwhile, when ran-dom matrix statistics hold, the mean value of r has beennumerically determined to be approximately 0.5295 ±0.000611.

In Figure 11, we present exact diagonalization resultsfor L = 12 lattices at half-filling with potential wavenum-

ber k = 1φ and the boundary conditions described in Sec-

tion II.C above. We show data for the same parameterrange examined in the body of this paper and averageover 50 samples for each value of g and u. For the largestvalue of u, the mean value of rn interpolates between theexpected values as g is raised, consistent with the exis-tence of a localization transition. We have also checkedthat the distributions of rn have the expected forms inthe small and large g limits in this regime. For smallervalues of u, we can speculate that 〈rn〉 grows with L atlarge g and approaches the expected value for very largeL. To argue for a MBL transition on the basis of exactdiagonalization, we would need to study this sharpeningof the crossover as L is raised. This would indeed be an

0 0.5 1 1.5

0.4

0.5

g

0 200 400 600 800 1000

0.35

0.4

0.45

0.5

0.55

g

00.040.160.320.64

FIG. 11: The mean of the ratio between adjacent gaps inthe spectrum, defined in (A4). This data was obtained byexact diagonalization of the parent model (1) for L = 12systems. All data points have been averaged over 50 samples,and the legend refers to different values of the interactionstrength u. The mean value of 〈rn〉 shows a crossover fromPoisson statistics (indicated by the bottom reference line) toWigner-Dyson statistics (indicated by the top reference line),for the largest values of u. Representative error bars havebeen included in the plot; the absent error bars have roughlythe same size.

interesting avenue for future work. For our present pur-poses however, we only want to check consistency withour real-time dynamics data, as we have done in Figure11.

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qualitatively from their counterparts in the non-interactingAA model. The non-interacting extended phase is not er-godic, indicating that interactions are necessary for ther-malization. Meanwhile, the many-body localized phaseis expected to exhibit logarithmic growth of the bipar-tite entanglement entropy to an extensive value, albeitwith subthermal entropy density. Such behavior is in factconsistent with the recent observations in the disorderedproblem13,19. This growth is absent in the AA localizedphase without interactions. Despite this difference, the in-teracting and non-interacting localized phases are similarin their inability to thermalize the particle density.

59 There is an exception to this statement: multi-photon pro-cesses do seem to play an important role deep in the er-godic phase, where the energy content of the system isespecially high. See Appendix A and the discussion of thetime-dependence of the autocorrelator χ in Section III.Afor more details.

60 To appropriately realize open boundary conditions, weshould also turn off interactions over the boundary. Whenexploring different options for the boundary conditions, wevaried J over the boundary and neglected to vary V . This isunfortunate in that it makes the model somewhat stranger.However, our boundary conditions are chosen for conve-nience anyway, and the numerics suggest that the choice ofboundary conditions does not impact the essential physicsdiscussed in this paper.

61 The statistical fluctuation of the total energy of the ran-domly chosen initial configuration is of order

√L. Suppose

the total energy is conserved by the dynamics. We canwrite E/

√L = x0 + hA0 cos θ0 = x∞ + hA∞ cos θ∞. Here,

the subscripts 0 and ∞ refer to the initial and late-timestates, x0 and x∞ are bounded random numbers capturingthe expectation value of interactions (and hopping at late-times), h is the non-random amplitude of the quasiperiodicpotential, and A0 and A∞ are positive bounded ampli-tudes of the Fourier components at the wavevector k of thequasiperiodic potential. This ansatz implies a finite correla-tion between the random phases θ0 and θ∞. Therefore, oneof the Fourier modes of χ remains correlated as L → ∞,

17

and we expect χ ∼ 1L

in the ergodic phase. Note that thisargument truly applies only to the energy-conserving par-ent model. In fact, in our numerics, there is only partialenergy conservation, and energy non-conserving events be-come more prevalent as u, g, or L is raised. This meansthat χ will generically decay faster than 1

Lat large L in

the ergodic phase.62 This statement should be interpreted with some care.

Quantum entanglement entropy measures, such as theRényi entropy that we define in equation (14), carry in-formation about the off-diagonal elements in the reduceddensity matrix. These terms have no classical analogue andwould not be considered in a thermodynamic calculation.This difference can result in discrepancies in the sublead-

ing behavior. For instance, consider our calculation of thebipartite Rényi entropy of the model state |Φ〉 in SectionIV.A: the quantum Rényi entropy is one bit lower thanthe Rényi entropy calculated by classical counting of con-figurations. A more precise analogue of the classical en-tropy would thus be a “diagonal” entropy in which alloff-diagonal elements of the reduced density matrix wereneglected.

63 Only the first term on the right-hand side of equation (22)would appear in a “classical counting” derivation of thethermodynamic entropy. The other two terms account foroff-diagonal elements in the reduced density matrix (20).Please see footnote 53 for more details.