-
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
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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-
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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.
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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
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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.
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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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phases differ
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
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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.