The Ergodic Side of the Many-Body Localization Transition

David J. Luitz1 and Yevgeny Bar Lev2

1Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA∗

2Department of Chemistry, Columbia University,3000 Broadway, New York, New York 10027, USA†

Recent studies point towards nontriviality of the ergodic phase in systems exhibiting many-bodylocalization (MBL), which shows subexponential relaxation of local observables, subdiffusive trans-port and sublinear spreading of the entanglement entropy. Here we review the dynamical propertiesof this phase and the available numerically exact and approximate methods for its study. We discussin which sense this phase could be considered ergodic and present possible phenomenological expla-nations of its dynamical properties. We close by analyzing to which extent the proposed explanationswere verified by numerical studies and present the open questions in this field.

CONTENTS

I. Introduction 1

II. Models 2

III. Properties of the Ergodic Phase 3A. Three flavors of ergodicity 3

1. Eigenvalue statistics 32. Eigenvector statistics, multifractality and

the “bad metal” 43. Eigenstate thermalization hypothesis 5

B. Entanglement Structure 6C. Dynamical Properties and Transport 6

1. Mean-square displacement and acconductivity 6

2. Autocorrelation function and theEdwards-Anderson parameter 8

3. Survival or return probability 84. Dynamical structure factor and the

imbalance 85. Entanglement entropy growth 96. Transport from nonequilibrium stationary

states 97. Summary 10

IV. Phenomenological explanations 11A. Griffiths effects 11B. Phenomenological Renormalization Group 13

V. Numerical Methods 14A. Exact methods for nonequilibrium time

evolution 141. Full diagonalization 142. Krylov space time evolution 143. tDMRG 144. tDMRG for open systems 155. Dynamical typicality 15

B. Exact methods for eigenstates calculation 161. Full diagonalization 162. Subset diagonalization 16

3. Excited state DMRG 164. Quantum Monte-Carlo 17

C. Approximate method: Perturbation theory 17

VI. Discussion and Open questions 18A. Subdiffusion and the subdiffusion to diffusion

transition 19B. Existence of the “bad metal” 19C. Mechanism of subdiffusive transport 20D. Relation to classical disordered models 20E. Summary 20

Acknowledgments 20

A. Relation between the mean-square displacementand the current-current correlation functions 21

References 21

I. INTRODUCTION

Boltzmann’s ergodic hypothesis — central to classicalstatistical mechanics — allows to derive most equilibriumresults. It states that a trajectory of a system with manydegrees of freedom will spend equal times in regions ofequal phase-space measure [1]. This implies that the in-finite time average of observables is equivalent to theirensemble average. Attempts to generalize this definitionof ergodicity to quantum systems had begun with theworks of von Neumann [2, 3] and substantial progress wasmade in the 1980ies both analytically and numerically inpioneering works by Berry, Pechukas, Peres, Feingold,Jensen and Shankar [4–12], culminating in the contribu-tions by Deutsch [13] and Srednicki [14–16]. It was real-ized early on that not all complex systems are ergodic,as in particular classically or quantum integrable systemsare nonergodic almost by definition. These systems arehowever not generic since integrability and thus noner-godicity is inherently unstable to the addition of genericperturbations [13]. Ergodicity breaking in more generic

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systems occurs during thermodynamic phase transitions,where a system spontaneously breaks a symmetry whenit orders [17]. A novel mechanism of ergodicity breakingin generic disordered quantum systems was proposed tenyears ago in a seminal work by Basko, Aleiner and Alt-shuler, a phenomenon now widely known as many-bodylocalization (MBL) [18]. This work established the stabil-ity of the nonergodic Anderson insulator to the additionof weak interactions at sufficiently small but finite en-ergy densities, and the stability of the (ergodic) metal forsufficiently large energy densities. It therefore predictedthe existence of a critical energy density (the so calledmany-body mobility edge) which demarcates the ergodicand the nonergodic phases. Unlike ergodicity breaking atthermodynamic transitions, this transition relies on thesystem being completely isolated from the environmentand has no signatures in static thermodynamic quanti-ties. The existence of a nonergodic phase was recentlyrigorously proved for one-dimensional random spin chainsunder a few physically reasonable assumptions [19, 20].Since MBL requires isolation from the environment, itsrealization in conventional condensed matter systems ischallenging [21, 22]. However signatures of MBL were ob-served in ultracold atomic gases on optical lattices bothin one-dimensional [23–25] and two-dimensional systems[26].

Most works on MBL concentrated on the study of thenonergodic phase, paying little attention to the ergodicphase [27–29]. The reason for this “injustice” is that fol-lowing the work of Basko, Aleiner and Altshuler it waslargely accepted that the ergodic phase in systems ex-hibiting the MBL transition is a trivial metal, namelyit has a finite dc conductivity [18]. Systems with un-bounded energy density, which were considered in thiswork, are essentially classical at sufficiently high energydensities and therefore have a finite dc conductivity ascan be shown using a self-consistency argument [18]. Forsystems with bounded energy density this is not the casesince even at infinite temperature there are examples ofsystems which are far from being classical [30]. First evi-dence of the nontriviality of the ergodic phase for systemswith bounded energy density was obtained by one of us[31]. Using a combination of nonequilibrium perturba-tion technique and exact diagonalization (see Section Vfor a brief description of these methods) a surprisinglyslow relaxation of the density autocorrelation functionwas observed on the ergodic side of the MBL transitionwhich was attributed to the existence of an intermedi-ate phase with impeded transport due to localized inclu-sions [31]. In a subsequent work, an extensive study ofspin transport in a large portion of the parameter phasespace was performed using exact diagonalization (ED)and the time-dependent density matrix renormalizationgroup (tDMRG) [32]. This study showed that most ofthe ergodic phase is subdiffusive up to simulated timesand argued that the dc conductivity must vanish if sub-

diffusion persists asymptotically in time. Similar resultswere obtained in the study of ac conductivity, where alsoa phenomenological explanation of the observed subdif-fusion was suggested [33].

In this review we concentrate on the ergodic phaseand refer the reader who is interested in the nonergodicphase or the MBL transition to Refs. [27–29] as also tomore recent reviews to appear in the current issue [34–38]. We limit the discussion to models with quencheddisorder and refer the reader interested in systems withquasiperiodic potentials to Ref. [39] for a review. Thestructure of the review is the following: in Sec. II wepresent the models which will be used throughout the re-view, in Sec. III C we survey the properties of the ergodicphase and discuss in which sense this phase is ergodic. InSec. IV we present the phenomenological theory of theergodic phase. Finally, we close the review by surveyingthe available numerical techniques in Sec. V and discussopen questions in Sec. VI.

II. MODELS

In this section we will introduce the models which willbe used throughout the rest of the review. Currently themost studied model in the context of many-body local-ization is the XXZ model,

H =Jxy2

L−1∑i=1

(S+i S−i+1 + S−i S

+i+1

)(1)

+ Jz

L−1∑i=1

Szi Szi+1 +

L∑i=1

hiSzi ,

were Szi , is the z−projection of the spin-1/2 operator,S±i , are the corresponding lowering and raising opera-tors, Jxy and Jz are inter-spin couplings and hi are ran-dom magnetic fields taken to be uniformly distributedin the interval hi ∈ [−W,W ] . This model conserves thez−projection of the total spin. Using the Jordan-Wignertransformation [40],

Szi → ni −1

2(2)

S+i → (−1)

∑i−1k=1 nk c†i

S−i → (−1)∑i−1k=1 nk ci,

it can be exactly mapped to a model of spinless electrons,

H = −tL−1∑i=1

(c†i ci+1 + c†i+1ci

)(3)

+ U

L−1∑i=1

(ni −

1

2

)(ni+1 −

1

2

)+

L∑i=1

hini,

3

where c†i creates a spinless fermion on site i and ni = c†i ciis the fermion density. We dropped a constant term andset t ≡ −Jxy/2 and U ≡ Jz to have a more conventionalnotation for fermions. The conservation of z−projectionof the total spin translates to the conservation of the totalcharge in the fermionic model. In most studies either thehopping t, or the in plane coupling, Jxy, are set to be one.We will pursue the latter convention here. Thus, unlessotherwise specified, all times are measured in units ofJ−1xy . For Jz = 1 (U/t = 2) both models have an ergodic

to nonergodic transition for a disorder strength of W =3.7±0.1 [41]. Since in this review we focus on the ergodicside of the transition, we mostly consider W ≤ 3.7 here.

Another model which we will discuss is the Anderson-Hubbard model,

H = −tL−1∑σ,i=1

(c†iσ ci+1,σ + c†i+1,σ ciσ

)(4)

+ U

L∑i=1

(ni↑ −

1

2

)(ni↓ −

1

2

)+

L∑i=1

hiσniσ,

where c†iσ creates a spinful fermion on site i and niσ =

c†iσ ciσ is the corresponding density. The disorder po-tential hiσ is taken to be different for the two speciesin order to explicitly break SU (2) symmetries in thecharge and the spin sectors thus avoiding possible com-plications [42, 43]. While this model naturally appears incold atoms experiments it is less popular than the XXZmodel, mostly because it has a larger local Hilbert spacedimension, which makes it more challenging for numeri-cal study.

Further models, which display an ergodic to noner-godic transition, are periodically driven systems. In thesemodels, also knowns as Floquet-MBL models, the tran-sition can be tuned by the frequency or the amplitudeof the drive. We have decided to exclude these systemsfrom our review due to scarcity of numerical results ontheir dynamics in the ergodic phase. A reader interestedin these topics is referred to the recent literature [44–52]and the review Ref. [37].

III. PROPERTIES OF THE ERGODIC PHASE

In this section we survey the numerical results on theproperties of the ergodic phase. Since the XXZ model (1)is equivalent to the spinless fermion model (3), in orderto avoid repetition we use the spin language in the restof the review. Readers who prefer to think in terms offermions are referred to the mapping (2). To minimizethe notational overhead we assume that all the consid-ered quantities are implicitly averaged over disorder re-alizations, and therefore are translationally invariant onaverage. In this section we will also uniformly use peri-odic boundary conditions and average over the volume

of the system, which we believe enhances readability andallows to operate with more physically transparent for-mulas. Readers who are interested in the technicalitiesand precise implementations are referred to Section V orto the original works.

A. Three flavors of ergodicity

We have postponed the precise definition of ergodic-ity which we use in this review to this subsection due tothe involved subtleties. While the definition for classicalsystems, via Boltzmann’s ergodic hypothesis, presentedin the beginning of the introduction is very precise, cur-rently, there is no commonly accepted definition of ergod-icity of quantum systems [10, 53]. Some of the reasonsfor this are that some concepts from classical physics like:microstates, phase-space and chaos cannot be immedi-ately carried over to quantum systems. Here, we discussthree different notions of ergodicity in quantum systemsbased on (i) the statistics of eigenvalues, (ii) the statis-tics of eigenvectors and (iii) the validity of the eigenstatethermalization hypothesis.

1. Eigenvalue statistics

For quantum systems which are chaotic in their clas-sical limit it was conjectured by Bohigas, Giannoni andSchmit that the eigenvalue statistics follow the statis-tics of an ensemble of random matrices, which dependson the symmetries of the Hamiltonian [54]. Using semi-classical field theory this conjecture was later justified[55]. For systems without a proper classical limit, suchas fermionic lattice models or spin systems, a direct con-nection between eigenvalue statistics and ergodicity isstill lacking. Nevertheless, it was empirically shown thatmany generic quantum systems do follow the eigenvaluestatistics of random matrices [56, 57]. To study the eigen-value statistics, the eigenvalues of the systems are calcu-lated and ordered ascendantly, then, traditionally, “un-folding” of the spectrum is performed, which eliminatesthe dependence of the statistics on the density of states.The distribution of the unfolded spacings is then obtainedand compared to the corresponding random matrix dis-tribution (Wigner-Dyson (WD) distribution). A systemis assumed to be ergodic if the distribution of its eigen-value spacing follows the WD distribution. The distribu-tion of eigenvalue spacings in disordered (Coulomb) inter-acting systems was studied a decade before MBL was es-tablished [58–61]. In these early studies a crossover froma Poisson to a WD distribution was observed. In laterstudies eigenvalue statistics for disordered spin chainswere also studied in the context of quantum chaos [62–64]. In the context of MBL eigenvalue statistics was firstconsidered in Ref. [65] which introduced a useful met-

4

ric for short-range correlations in the eigenvalues statis-tics, effectively eliminating the arbitrariness which existsin the unfolding procedure [66]. Instead of unfolding,the eigenvalue spacings δn = En+1 − En (where Enarethe ordered eigenvalues) are normalized by their mag-nitude, rn = min (δn/δn+1, δn+1/δn) [67]. Ergodicity isassumed when the obtained probability distribution of rn(or the unfolded δn) matches the one of the correspondingrandom matrix ensemble [68]. The fact that the phase,which is the subject of this review, is ergodic in this sensewas first established in Ref. [67] and then repeatedly inalmost every work on MBL.

The distribution of the eigenvalue spacings can beviewed as a stationary distribution of a Brownian motionin a space of Hamiltonians, where at each step a differentdisorder realization is drawn [69, 70]. In this approach,commonly known as the effective plasma model, unfoldedeigenvalues are thought of as particles with an effectivetwo-body interaction which is responsible for the eigen-value repulsion. By noting that in a second order expan-sion in the disordered potential the effective interaction iswell described by a power law, Serbyn and Moore derivethe corresponding limiting spacings distribution,

P (δn) = C1δβn exp

(−C2δ

2−γn

), (5)

where C1,2 are constants, 0 ≤ β ≤ 1 controls the levelrepulsion and γ controls the tail of the distribution [71].This distribution interpolates between the Poisson dis-tribution γ = 1, β = 0 and the WD distribution γ = 0,β = 1. Motivated by this form Serbyn and Moore numer-ically obtain γ within the whole ergodic phase, even out-side the region of validity of the effective plasma model(W ? 2). It is argued that while for weak disorderW . 2the spacings distribution appears to flow to the WD dis-tribution, for stronger disorder close to the MBL transi-tion 2 .W . 3.7 a region with intermediate statistics isfound. The corresponding distribution is similar to thecritical distributions obtained for Anderson transitions, ithas an exponential tail γ = 1 and a finite level repulsionβ > 0 [72]. It is therefore argued that the MBL transi-tion has critical statistics similar to the critical Andersonstatistics [73]. The effective model used by Serbyn andMoore was criticized in a follow-up exact diagonalizationwork [74], which pointed out that the eigenvalue statisticsof the ergodic phase does not appear to be scale invariantas the plasma model of Ref. [71] suggests, moreover thecritical eigenvalue statistics seems to better agree witha Poisson distribution, similarly to critical statistics ofAnderson transition on a Bethe lattice.

2. Eigenvector statistics, multifractality and the “bad metal”

The first proposal of an intermediate phase sandwichedbetween the deeply ergodic and nonergodic (MBL)

phases appeared almost 20 years ago [75]. This phase,colloquially dubbed by Altshuler a “bad metal” [76], wasfirst defined as a delocalized yet nonergodic phase. Thedefinition of ergodicity and delocalization in this contextis however quite different from what we have discussedabove, therefore to avoid confusion we will use a sans seriffont face to designate this kind of ergodicity.

The motivation behind this definition is best under-stood for the case of a single particle. The moments ofthe eigenstates of the single particle Hamiltonian ψα (x)written in the position basis are given by,

Iαq =∑x

|ψα (x)|2q . (6)

Delocalized single-particle eigenstates (for example eigen-vectors of a random matrix) scale as ψα (x) ∼ V −1/2

where V is the volume of the system, which yields Iαq ∝V −(q−1). Localized eigenstates which decay exponen-tially with distance from some localization center, yieldIαq ≈ const. Since the infinite time average of the densityautocorrelation function is given by Iα2 (see derivationin Eq. (24)), a natural definition of a delocalized (local-ized) state would be a state with Iα2 → 0

(Iαq → const

)for V → ∞. The participation ratio 1/Iα2 quantifies thenumber of sites that a eigenstate occupies in real space.When this number of sites is extensive, the system is de-fined to be ergodic. On the contrary when eigenstatescover a subextensive volume in real space Iα2 ∼ V −D2

with 0 < D2 < 1, not all sites in real space are “available”and the system is therefore nonergodic. The eigenstatewill be called multifractal if the generalized dimensionsDq depend on q. It occurs, for example, at the criticalpoint of the Anderson transition, where all the momentsIαq follow an anomalous scaling Iαq ∼ V −Dq(q−1) [72]. Westress that the sparseness of the eigenstates in real spacedoes not imply that a generic initial condition will belocked to a region in space. In fact almost all initial con-ditions will explore the whole volume of the lattice. Thesparseness of the eigenfunctions in real space has impli-cations on the dynamics of the wavepackets, which willbe subdiffusive with a dynamical exponent which couldbe related to the generalized dimension D2 [77, 78].

The many-body problem is equivalent to a single-particle hopping on a complicated graph, where the nodesof the graph represent many-body states weighted by thediagonal part of the Hamiltonian and the hopping ratesare given by the offdiagonal part. The apparent simplic-ity of this view is however misleading, since the disorder(many times taken to sit on the diagonal part) will behighly correlated. The number of return paths on thisgraph is exponentially small in their length, therefore byneglecting the loops it could be approximately mappedto a Cayley tree [75] (see also review by Imbrie et al.[34]). Using this analogy one can carry over the abovedefinition of ergodicity to the many-body case by substi-tution of the volume in real space by the total number of

5

many-body states, N ,

Iαq =∑n

|〈α|n〉|2q ∝ N−Dq(q−1), (7)

where |α〉 are the eigenstates of the Hamiltonian com-puted in some basis |n〉. We note in passing that thisquantity is closely related to the basis dependent Rényi“participation” entropies of the wave function as consid-ered in [41, 79, 80],

SP,αq =1

1− qln Iαq ∝ Dq lnN . (8)

In the limit of q → 1 it reduces to the Shannon entropySP,α1 = −

∑n |〈α|n〉|2 ln |〈α|n〉|2, and allows to define D1

as of Sα1 / lnN . There are a few problems with this def-inition of ergodicity. First, while the real space basis isa natural choice for the single particle problem there isno obvious choice of the basis |n〉 in the many-body case.The second and more serious problem is the lack of a di-rect connection between the spreading of the wavepacketon a complicated graph or tree in the many-body Hilbertspace and the dynamics in real space (cf. Eq. (23) for onepossible connection). In particular, it is not clear whetherthe sparseness of the eigenfunctions in the many-bodyspace (Dq < 1) has implications on the thermalizationin finite many-body systems, or has a signature in localobservables (for a discussion see Ref. [81]).

The existence of a stable delocalized but nonergodicphase was tested in numerous numerical studies. Moststudies of multifractality are focused on either the Bethelattice or random-regular graphs. After almost a decadeof study, this question is still largely open [82–90], whilemost extensive numerical studies suggest that this phasedisappears in the thermodynamic limit [87, 89]. For phys-ical lattice models this question was considered in a studyof a random Josephson array [91] and for the XXZ model[41, 92, 93], with a similar inconclusive outcome. WhileRef. [41] suggests that D1 = 1 below the MBL transi-tion, Refs. [91, 93] argue in favor of a stable intermediatephase with D2 < 1. Furthermore, Ref. [92] argues thatthis phase shrinks to the MBL critical point in the ther-modynamic limit. We would like to point out that itis possible that this apparent discrepancy might followfrom a different basis used to calculate Iαq in these works[41, 92, 93].

An attempt to connect the notion of ergodicity fromeigenvector statistics to ergodicity defined through eigen-values statistics was performed by Serbyn and Moore inthe work described above [71]. By using a specific choiceof the basis

∣∣ψβ⟩ ≡ 2Szi |β〉 (where |β〉 are the eigenstatesof the Hamiltonian) in (7) one can write,

Iαq =∑β

∣∣⟨α|ψβ⟩∣∣2q ∝ N−Dq(q−1), (9)

where the scaling with the size of Hilbert space is takenas an assumption (only Iα2 was considered in Ref. [71]).A similar scaling of the moments, was conjectured inRef. [94] and was recently numerically verified [92]. Usingheuristic arguments, Serbyn and Moore connected theexponent γ in (5), which parametrizes the distributionof the eigenvalue spacing to the generalized dimensionγ = 1−D2 [95]. This relation was however never verifiednumerically.

3. Eigenstate thermalization hypothesis

In this review we utilize yet another definition of er-godicity, which is more similar to the Boltzmann ergodichypothesis for classical systems [1] as also to ideas by vonNeumann [2, 3]. It is commonly known as the eigenstatethermalization hypothesis (ETH) and it was mostly de-veloped by Deutsch and Srednicki more than two decadesago [13–16, 96], based on a multitude of theoretical andnumerical works in quantum chaos [5–12] (for recent re-views, see [81, 97, 98]). One can show that a sufficientcondition for local quantum observables O to decay totheir microcanonical value (and to stay close to this valuefor sufficiently long times) is the validity of the ansatz,⟨

α∣∣∣O∣∣∣β⟩ = O (E) δαβ + e−S(E)/2f (E,ω)Rαβ , (10)

where |α〉, |β〉 are eigenstates of the Hamiltonian, S (E)is the microcanonical entropy, O (E), f (E,ω) are smoothfunctions of their arguments with E ≡ (Eα + Eβ) /2 andω = Eβ−Eα and Rαβ are random independent variableswith zero mean and a unit variance.

The fact that the ergodic phase is indeed ergodic un-der this definition was established for the diagonal ele-ments in Refs. [99, 100] and for the offdiagonal elementsin Ref. [101]. However it was observed that the shapeof the probability distributions of local operators in theeigenbasis of the Hamiltonian according to Eq. (10) de-pends strongly on the value of the disorder strength forboth diagonal and off-diagonal matrix elements. In par-ticular, the distributions are perfectly Gaussian for weakdisorder, while for intermediate disorder strength the dis-tribution of Rαβ becomes strongly non-Gaussian even inthe thermodynamic limit. Interestingly, even in this case,the ETH ansatz remains valid, although in a generalizedform with a non-Gaussian noise term Rαβ . These non-Gaussian probability distributions are accompanied witha slower decrease of the standard deviation of the offdiag-onal matrix elements

⟨α∣∣∣O∣∣∣β⟩ with the size of the sys-

tem in the low frequency limit (ω = Eα−Eβ). This mod-ified scaling was connected to the dynamical exponent ofthe system [101]. The dependence of the

⟨α∣∣∣O∣∣∣β⟩ ma-

trix elements on ω was also studied in Ref. [92].

6

B. Entanglement Structure

The ETH ansatz is commonly assumed to hold for few-body operators which have a finite support on a smallsubsystem of the total isolated system. This implies thateven when the whole system is in an eigenstate |α〉, asufficiently small subsystem A is thermalized by the restof the system. Clearly, thermalization requires that theentropy of the subsystem, i.e. the von Neumann entan-glement entropy obtained by tracing out degrees of free-dom that are not in the subsystem A, (cf. also the recentreview in Ref. [102])

SαA = −Tr (ρA ln ρA) , with ρA = TrA|α〉〈α|

has to be extensive in the subsystem size SαA ∝ LA, whichis usually referred to as a volume law scaling. In the MBLphase, on the other hand, this is not true and due tothe finite localization length, the entanglement entropy ofMBL eigenstates scales as the surface area of the subsys-tem (which is constant in one dimension). This differencein the entanglement scaling across the MBL transitionwas first observed by Bauer and Nayak [103] numericallyand has subsequently become a popular measure to de-tect the MBL transition [41, 104–107].

Kjäll et al. numerically studied the critical region inwhich the dominant scaling changes from a volume lawto an area law [104]. They discovered that close to thetransition the variance of the entanglement entropy ex-hibits a maximum. A careful analysis of the probabilitydistributions of the entanglement entropy showed thatclose to the transition, a mixture of volume-law and area-law states exists [100, 107]. In Ref. [107], it was shownthat in periodic, disordered, one dimensional systems theaverage of the entanglement entropy SA over all pos-sible bipartitions with the same subsystem length is asmooth and concave function of the subsystem lengthLA. The derivative ∂SA/∂LA was argued to capture thedominant entanglement scaling in the system for singleeigenstates, and is close to its maximal value (ln 2 forspin-1/2 chains) in the case of a volume law scaling, andzero in the case of an area law scaling. The probabil-ity distribution of ∂SA/∂LA close to the MBL transitionbecomes strongly bimodal even for single disorder real-izations [107], although the inter-sample variance is ob-served to be smaller than the sample-to-sample variance[107, 108].

At weaker disorder, the probability distributions of theentanglement entropy and its slope are sharply peaked ata large, volume law value, with exponentially suppressedtails at lower entanglement, which neither affect the meannor the variance. The analysis of the entanglement struc-ture in Ref. [107] seems to exclude the possibility of crit-ical eigenvectors which have a volume law scaling with asuppressed prefactor [109]. Such a sub-thermal volume

law scaling holds only for the disorder averaged entangle-ment entropy and is caused by the disorder average overa sharply peaked bimodal distribution. This average cor-responds to a part of the distribution with exponentiallylow weight and is therefore physically meaningless.

The study of the spatial entanglement structure is es-pecially interesting in light of the rare Griffiths regionspicture, which was proposed to explain the observed sub-diffusion (see Sec. IV). In Ref. [100], the entanglemententropy was calculated as a function of the cut position,showing qualitatively that in the ergodic phase some cutsbetween the two subsystems have much lower entangle-ment entropies compared to other cuts. Moreover theseregions can be identified in any eigenstate of the system.The correlation between eigenstates in the spatial vari-ation of entanglement was also observed in the nearestneighbor concurrence as a local probe of entanglement[110]. In an analysis of the probability distribution of thechange of the entanglement entropy ∆(`) = S(`+1)−S(`)if the subsystem is enlarged by one site it was shown thatat intermediate disorder there is an increasing probability(when disorder is increased) of finding ∆(`) < 0, whichcan be seen as indirect evidence for the existence of insu-lating inclusions in the system [107]. To make progressin this direction, it may prove useful to study more localprobes of entanglement, such as the mutual informationI(A,B) = SA+SB−SA∪B , which was recently proposedas a generic measure to extract the correlation length[111].

C. Dynamical Properties and Transport

In this section we will survey the different results onthe dynamical properties of the ergodic phase of systemsexhibiting MBL transition. To emphasize the similaritybetween MBL systems and classical glasses through thissection we will adopt the notation commonly used in theglasses community. At the end of the section we present asummary of the relations between the different dynamicalquantities.

1. Mean-square displacement and ac conductivity

The XXZ model conserves both the total energy andthe z−projection of the total spin, which is equivalentto the conservation of the total number of particles inthe fermion language. For this model one can thereforestudy either the transport of spin or energy. Followingthe work of Basko, Aleiner and Altshuler [18] and firstnumerical studies of the dc conductivity it was largely be-lieved that the ergodic phase is a metal [112–114], namelythat it has a finite dc conductivity (similarly to a normalliquids in structural glasses). First evidence of the sur-prisingly slow relaxation of local observables deep in the

7

ergodic phase was obtained using a self-consistent secondBorn approximation [31]. A more extensive explorationof transport in the ergodic phase using numerically exactmethods was performed in Ref. [32]. In this work thespin-spin correlation function

Gr (t) = Re1

L

∑i

Tr[ρ0S

zi+r (t) Szi (0)

](11)

was calculated, where ρ0 is the density operator of theinitial state of the system which is typically taken to beproportional to the identity operator (infinite tempera-ture). This correlation function is analogous to the vanHove correlation function in structural glasses [115], andintuitively describes the evolution of a spin excitationcreated at time t = 0. To assess transport propertiesone can evaluate the analog of the classical mean-squaredisplacement,

x2 (t) =∑r

r2Gr (t) , (12)

which for diffusive systems should asymptotically scalelinearly with time, x2 ∼ t. It was found that even for thesmallest studied disorder strength (W ≈ 1) and throughmost of the ergodic phase transport is subdiffusive,

x2 (t) ∼ t2/z for t < t∗ (Jz,W ) , (13)

with a dynamical exponent, z (Jz,W ) ≥ 2 which de-pends on the parameters of the system. The simulationtime t∗ ≈ Lz was chosen such that finite-size effects wereeliminated up to a predefined precision [32]. This timescale could be considered as a generalized Thouless time,namely the time it takes to transport a particle across thesystem [74, 92, 101, 116]. An analogous calculation wasperformed using the self-consistent second Born approxi-mation for a two-dimensional Anderson-Hubbard model(4), yielding similar results [117]. The evaluation of (11)and (13) is valid for any initial state and therefore doesnot require the system to be within the linear responseregime. However for a thermal initial state ρ0 it is di-rectly related to the frequency dependent diffusion coef-ficient calculated from linear response theory [118] (for aderivation of this relation for quantum systems see Ap-pendix A),

D (ω) = −ω2

∫ ∞0

dt eiωtx2 (t) , (14)

which is proportional to the ac conductivity,

σ (ω) ∝ ω2

∫ ∞−∞

dt eiωt|t|2/z ∝ |ω|1−2/z. (15)

The dependence of the ac conductivity on the frequencywas numerically calculated in Refs. [33, 119]. We notein passing that for infinite temperatures what is actually

computed is D (ω) ∼ Tσ (ω) since σ (ω) vanishes in thislimit. At the MBL transition the dynamical exponent isexpected to diverge, z → ∞, and therefore the criticalac conductivity is σ (ω) ∝ ω [119]. The ac conductivitywas the first dynamical quantity which was studied inthe context of MBL. Within the linear response theoryits real part is given by the Kubo formula [120],

Re σ (ω) =1

ωLtanh

(1

2βω

)∫ ∞−∞

dt eiωtRe⟨J (t) J (0)

⟩β,

(16)where β = 1/T , and we set the Boltzmann constant tobe one, 〈.〉β is the thermal expectation value and J is thetotal current density operator,

J = iJxy2

∑n

(S+n S−n+1 − S

+n+1S

−n

). (17)

The use of the Kubo formula above assumes the valid-ity of linear response theory. While the validity of (16)within the ergodic phase was not directly tested, the re-sponse of the system for sufficiently small driving fieldswas shown to be linear [121, 122], as also the heating ofthe system [123].

The first results on spin and heat ac conductivities wereobtained using exact diagonalization (ED) [112]. In thiswork it was argued that,

σ (ω) w σdc (Jz,W ) +A |ω| (18)

(and similarly for the heat conductivity) with σdc > 0 formost Jz and W . The putative delocalization of the MBLphase was later challenged in Ref. [113] and then also inRef. [114]. Recent large scale ED studies, with systemsas large as L = 28, confirmed a linear scaling with fre-quency and a finite dc conductivity in the ergodic phase[124–126], in contrast to the finding of a sublinear scal-ing (15) and zero dc conductivity in the same region ofthe phase diagram as argued in Refs. [33, 119]. Thesecontradicting results highlight the difficulty of extractingthe low frequency behavior from the Kubo formula (16);a difficulty, which was pointed out already by Thoulessand Kirkpatrick [127] and more recently by Berkelbachand Reichman [113]. The evaluation of the ac conduc-tivity for any finite system requires a broadening of themany-body levels with an artificial width η, which couldbe attributed to either a residual coupling to the envi-ronment or to the timescale over which the conductivityis measured [124]. This coupling results in σdc (η) > 0 forany finite system. In order to eliminate the dependenceon η it is crucial to take the thermodynamic limit L→∞before taking η → 0+ [127, 128]. For systems known tobe metallic (σdc > 0), this apparently formidable task isactually feasible, since even for finite systems σ (ω) is al-most independent of η, as long as η > ∆ (where ∆ is themean level spacing) [127]. This is however not the casewhen it is not known a priori if σdc > 0, and the way the

8

extrapolation to the thermodynamic limit is performed isextremely important. The main technical difference be-tween Refs. [112, 114, 124–126] and Refs. [33, 119] is thefunctional form which was used to fit the ac conductivity.While the former works assume a finite dc conductivityand the form (18), the later assume that the dc conduc-tivity vanishes and the form (15). An attempt to cir-cumvent the inherent finite size constraint of ED studieswas performed in Ref. [129], where a continued fractionexpansion of dynamical correlations using a variationalextrapolation of recurrents was developed. This allowedthe authors of Ref. [129] to work essentially at the infi-nite system limit. The results of this work are consistentwith a vanishing dc conductivity and the functional de-pendence (15).

2. Autocorrelation function and the Edwards-Andersonparameter

A different way of examining dynamical propertiesis the calculation of the local autocorrelation function,which is a special case of zero displacement (r = 0) in(11),

G0 (t) =1

LRe

L∑i=1

Tr ρ0Szi (t) Szi (0) . (19)

Its infinite time average for thermal initial states, ρ0 =

exp[−βH

]/Z is given by,

limT→∞

∫ T

0

G0 (t) dt =1

L

∑i

∑α

e−βEα∣∣∣⟨α ∣∣∣Szi ∣∣∣α⟩∣∣∣2 = qEA,

(20)where |α〉 and Eα are the eigenvectors and eigenvaluesof the Hamiltonian and qEA is the Edwards-Anderson(EA) parameter [130]. Similarly to the situation for spin-glasses the EA parameter is zero in the ergodic phaseand nonzero in the nonergodic MBL phase, and couldbe used as an order parameter of the MBL transition[131][104, 132–135]. While there is no direct connectionbetween the decay of the autocorrelation function andtransport, many times the following relation between theautocorrelation function and the mean-square displace-ment is assumed to hold (see derivation for subdiffusiveclassical systems in Sec. IV),

G0 (t) ∝ 1√x2 (t)

= t−1/z, (21)

which relies on a scaling hypothesis, and allows to re-late between the exponents of the ac conductivity (15)and the autocorrelation function (21) [136]. This rela-tion was also derived in Refs. [33, 119]. The spectral

density can be evaluated by taking the Fourier transformof the autocorrelation function,

A (ω) ≡∫

dt eiωtG0 (t) ∝ |ω|−(1−1/z), (22)

which diverges at small frequency [137]. While the auto-correlation function was already considered in Refs. [113,138], the surprisingly slow relaxation deep in the ergodicphase was noted in Ref. [31], and was attributed to thepossibility of an intermediate phase. In fact, a directstudy of the functional dependence of the dynamical ex-ponent extracted from the autocorrelation function (21)was only performed quite recently [33, 101].

3. Survival or return probability

For an initial state which is a projector on an eigen-state, ρ0 = |α〉 〈α| the autocorrelation function is closelyrelated to the survival probability,

C (t) =∣∣∣⟨α ∣∣∣δSzi (t) δSzi (0)

∣∣∣α⟩∣∣∣2 =∣∣∣⟨ψα ∣∣∣e−iHt

∣∣∣ψα⟩∣∣∣2 ,(23)

where we have defined, |ψα〉 ≡ δSzi |α〉, and δSzi =

Szi −⟨α∣∣∣Szi ∣∣∣α⟩, to set the infinite time average of⟨

α∣∣∣δSzi (t) δSzi (0)

∣∣∣α⟩ to zero. The decay of the sur-vival probability therefore corresponds to the decay ofthe fluctuations of the autocorrelation function. The in-finite time average of the survival probability is given by,

I2 ≡ limT→∞

1

T

∫ T

0

dt

∣∣∣∣∣∣∑β

|Cαβ |2 e−iEβt

∣∣∣∣∣∣2

=∑β

|Cαβ |4 ,

(24)where we defined Cαβ =

⟨α∣∣∣δSzi ∣∣∣β⟩ and I2 is the inverse

participation ratio. The inverse participation ratio scalesas I2 ∝ N−D2 , where N is the Hilbert space dimensionand D2 is a generalized dimension. For delocalized sys-tems (even non-interacting) D2 > 0 and I2 → 0 in thelimit L → ∞, while for localized systems, D2 = 0 andI2 → const. There is no direct connection between thedecay of the survival probability in many-body systemsand transport, yet a power law relaxation was obtained[139],

C (t) ∼ t−D2 , (25)

with disorder dependent exponent, 0 ≤ D2 ≤ 1, which isjust the generalized dimension defined above [93, 140].

4. Dynamical structure factor and the imbalance

Instead of studying the decay of local excitations onecan also consider the relaxation of collective (spin-wave

9

like) excitations,

F (q, t) = Re Tr[ρ0S

zq (t) Sz−q (0)

], (26)

where Szq (t) =∑n S

zn (t) exp [iqn] /

√L, and F (q, t) is

the analog of the coherent intermediate scattering func-tion in structural glasses [115]. It is simply relatedto the van Hove correlation function (11) calculated inRefs. [32, 117],

F (q, t) =∑r

G (r, t) e−iqt, (27)

and was directly studied in the context of MBL inRef. [141]. For diffusive systems this quantity re-laxes exponentially, F (q, t) ∼ exp

[−Dq2t

], while for

supercooled liquids the relaxation is characterized bya Kohlrausch-Williams-Watts (KWW) law, F (q, t) ∼exp

[−Atβ

], and is also known as the β−relaxation [115].

Taking a Fourier transform of F (q, t) with respect to timegives the dynamical structure factor S (q, ω) which wasrecently numerically studied in Ref. [126].

Due to the destructive nature of measurements in coldatoms experiments it is hard to measure two-time cor-relation functions, however for q = π and a Néel stateinitial condition Eq. (26) reduces to a one time-quantity,dubbed the imbalance,

I (t) =

L∑n=1

(−1)n⟨Szn (t)

⟩, (28)

which was successfully measured in a cold atoms exper-iment [23]. An extensive ED study of the decay of theimbalance (generalized to random product states) wasperformed by one of us in Ref. [142], where it was foundthat for systems up to L ≤ 28 the imbalance decays as apower-law superimposed on decaying oscillations,

I (t) ∼ t−ζ(W ), (29)

moreover the dynamical exponent is subdiffusive,ζ (W ) < 1/2 for disorder strengthsW > 0.5 and vanishesat the MBL transition. The exponent ζ (W ) is related tothe dynamical exponent as ζ (W ) = 1/z [143][101].

5. Entanglement entropy growth

In Ref. [142] it was also demonstrated that after a localquench the entanglement entropy grows only sublinearlywith time,

S (t) ∼ t1/zent(W ), (30)

such that zent (W ) ≥ 1. Similar results were obtainedusing a light-cone tDMRG, which allows to obtain bulktransport up to some finite time [144]. This study used a

binary disorder distribution which allowed to exactly av-erage over all disorder realizations by utilizing the ancillatrick [145].

If one assumes that “quasi-particles” become entan-gled on “first encounter” and cannot disentangle, thenit is clear that entanglement has to spread faster thantransport of particles, zent < z [146]. Indeed, due tothe conservation of the total spin, in order to reducethe total spin in some interval l, one has to transporta “quasi-particle” through the interval l times. On theother hand, by the assumption above, to entangle all the“quasi-particles” in this interval, it is enough to transporta “quasi-particle” through it only once. This implies thatthe time it takes to induce entanglement in this intervalis l times smaller than the transport time, tent = ttr/l or,

zent = z − 1. (31)

This heuristic argument establishing the connection be-tween the dynamical exponents z and zent was introducedin Refs. [105, 106]. While a microscopic derivation ofthe dynamical exponent zent and its connection to thedynamical exponent z is still missing, using a novel di-agrammatic technique a related quantity was calculatedby Aleiner et al. [147]. In this work an equation of motionfor the out-of-time order correlator was obtained whichis similar to equations of motion customary in the fieldof combustion. The out-of-time order correlator mea-sures the spread of disturbances [148] and is related tothe spread of entanglement [149]. Ref. [147] providestherefore a microscopic justification to the “entanglementon first encounter” conjecture raised by Kim and Huse[146]. For a numerical verification of this conjecture seeRef. [150]. Finally, we note that slow information trans-port was also observed in a sublinear power law growthof the operator entropy of the time evolution operator[151].

6. Transport from nonequilibrium stationary states

Another initial condition which is useful for coldatoms experiments is the domain wall initial condition,|↑ · · · ↑↓ · · · ↓〉 , where one measures the decay of the mag-netization imbalance between two halves of the system[26]. While for this initial condition dynamics has notyet been studied in an experiment, it was simulated us-ing tDMRG for systems up to L = 60 in Ref. [152]. Thetransported magnetization across the domain wall is con-sistent with a power law in accord with the scaling (13),M (t) ∼ t1/z [153]. The domain wall initial condition andthe decay of the longest wavelength excitation were alsoused in an ED study of energy transport [154]. In thiswork, using a phenomenological diffusion equation andby extrapolating to the thermodynamic limit, an energydiffusion coefficient was calculated. It was argued that

10

energy diffusion coefficient is nonzero through a largeportion of the ergodic phase [154].

In condensed matter systems, where the real time dy-namics is fast, and therefore mostly inaccessible, trans-port is normally assessed by the calculation of a station-ary state current after the system has been connectedto a constant bias. Normal diffusive metals obey Ohm’slaw with a stationary current which decreases as L−1.For ballistic metals (with mean-free path larger than thesystem size) the current does not depend on the size ofthe system and for (perfect) insulators the current de-creases exponentially with system size. More generally arelation between the dynamical exponent and the decayof the current can be established using the following clas-sical consideration [155]: The time it takes for one spin tobe transported from one side of the system to the otheris given by t∗ = Lz, (which is the generalized Thoulesstime defined below Eq. (13)). Since a fixed bias makes anextensive number of spins available for transport, N ∝ L,the stationary current is given by the ratio,

j ∝ L

t∗=

1

Lz−1. (32)

A power law dependence of the stationary current onthe system size was obtained in an open system tDMRGstudy of the ergodic phase [122]. A direct comparison tothe dynamical exponent was not performed there, but itwas found that for W > 0.5 spin transport in the systemis subdiffusive, while for W < 0.5 it appears to be dif-fusive (see Section VA for a description of the method).On the right panel of Fig. 1 we present the dynamical ex-ponent 1/z calculated from equation (32) and using thedata of Ref. [122]. To highlight the importance of finitesize effects we plot the same data, restricting the systemsizes to L < 100. For W < 0.6 and small system sizes(L < 100) the transport appears to be faster than diffu-sive, while for larger sizes (L < 400) the transport slowsdown, yet remaining slightly faster than diffusive, evenfor the largest system sizes which were used in Ref. [122].To estimate the minimal system sizes for which the ef-fects of the disorder become important, Žnidarič et al.calculate the mean-free path in the system in second or-der perturbation theory in the weak disorder. This lengthscales as l ∝W−4/3[156], and forW < 0.6 becomes largerl L then the system sizes available in ED, making EDan inappropriate numerical tool for the study of such asmall disorder. While the results of Ref. [122] are consis-tent with asymptotic diffusion for W < 0.6, whether foreven larger system sizes transport slows down and even-tually becomes subdiffusive is still an open question. Ifthis is indeed the case it will suggest that another lengthscale l (W,U) > l exists in this problem.

7. Summary

For the convenience of the reader we summarize all theresults presented in this subsection,

x2 (t) ∼ t2/z G0 (t) ∼ I (t) ∼ t−1/z

σ (ω) ∼ ω1−2/z A (ω) ∼ ω−(1−1/z)

C (t) ∼ t−D2 I2 ∼ N−D2

j (L) ∼ L−(z−1) S (t) ∼ t1/(z−1) (33)

where 2 ≤ z < ∞ is the dynamical exponent, the mean-square displacement x2 (t) is defined in (13), the auto-correlation function G0 (t) in (21), the imbalance I (t)in (29), the spectral density A (ω) in (22), the survivalprobability C (t) in (23), the inverse-participation ratioI2 in (24) and S (t) is the entanglement entropy. Here wehave considered only disorder averaged quantities. Forthe discussion of the corresponding typical quantities werefer the reader to Ref. [137].

In Fig. 1 we compare some of the exponent relationswhich where discussed in this section and are summa-rized above. We are skipping comparisons of trivial rela-tions which follow from a Fourier transform, such as thecomparison between the exponents of G0 (t) and A (ω).One of the most commonly used and assumed relationsis the relation between the decay of the autocorrelationfunction and the growth of the mean-square displacementG0 (t) ∼

⟨x2 (t)

⟩−1/2 [see Eq. (21)]. While this equationclearly holds for diffusive transport since the excitationprofile Gr (t) is asymptotically Gaussian, there is no rea-son why it should a priori hold for subdiffusive systemswhere asymptotic excitation profiles can have heavy tails.This relation was indirectly tested in Refs. [33, 129] yield-ing not a very compelling agreement. In the left panel weperform a direct test of this relation using the dynamicalexponent 1/z obtained from the decay of the autocorre-lation function in three different studies [33, 101, 129],compared to the dynamical exponent computed from thegrowth of the mean-square displacement [32]. We notethat while the results across the studies do agree qualita-tively the quantitative discrepancy is pretty large, some-times as large as 100%. We attribute this discrepancy tothe difficulty of fitting power laws to data on a limitedtime domain and with superimposed oscillations, notingthat the growth of the mean-square displacement doesnot seem susceptible to such problems. Another possi-ble resolution could be that the extracted power-laws arenon asymptotic with different measures having differentsensitivity to the finite size effects. The relation betweenthe two exponents seems to hold well for W < 2 with anincreasing discrepancy for stronger disorder, however thedifficulty of reliably extracting the autocorrelation expo-nent precludes from drawing strong conclusions. On theright panel of Fig. 1 we compare the relation betweenthe transport dynamical exponents extracted from the

11

mean-square displacement and the dynamical exponentobtained from the ac conductivity, entanglement entropyand the decay of the stationary current. Due to rela-tion (40) the ac conductivity exponents and the mean-square exponents have to agree asymptotically, which isindeed what we observe. The exponent extracted fromeffectively infinite systems is however dramatically dif-ferent [129]. To the best of our knowledge relation (31)was never explicitly verified for disordered systems. Toverify this scaling we plot the transport dynamical ex-ponent 1/z and 1/ (zent + 1) as obtained in Ref. [142] onthe right panel of Fig. 1. It is clear that the relationholds only qualitatively with increasing discrepancy forstronger disorder. Since finite size effects are negligiblefor strong disorder it appears that the relation betweenthe exponents is more intricate than what is suggested byEq. (31). Interestingly, while there is a clear violation ofthe relation (32) the exponent extracted from the currentcoincides with the exponent extracted from entanglementgrowth. We note in passing that since entanglement can-not spread faster than particles there is an upper boundof 1/2 on the value of the 1/z exponent extracted fromentanglement growth. This means that the agreement forW < 1 is in some sense trivial, moreover in this regimeas was pointed out in the end of Sec. III C 6, ED resultsbecome increasingly unreliable for such a weak disorderdue to severe finite size effects. The apparent violation ofrelation (32) for strong disorder, where finite size effectsare not pronounced has to be better understood.

IV. PHENOMENOLOGICAL EXPLANATIONS

A. Griffiths effects

Anomalous diffusion and subexponential relaxation ofautocorrelation functions are often associated with a fail-ure of the central limit theorem and the presence of heavytailed distributions [157–159]. For example, for classi-cal spin glasses a broad distribution of relaxation timesyields a subexponential relaxation of the magnetizationand spin autocorrelation functions [160–163], and in thecase of Lévy’s flights, superdiffusion is a result of a broaddistribution of the hopping distances [164]. A broad dis-tribution of relaxation times was also proposed as an ex-planation for the observed subdiffusion in ergodic one-dimensional systems exhibiting MBL [33]. Microscopi-cally the “fat tail” of the distribution of the relaxationtimes follows from exponentially rare inclusions whichhave exponentially long relaxation times and thereforeyield non-negligible contributions. Rare region effects onthermodynamical phase transitions were first studied byGriffiths, who noted that quenched disorder can make thefree energy non-analytic in a finite temperature interval[165]. The importance of rare spatial regions in quantumphase transitions and for dynamical properties is even

0 1 2 3 4

W

0.0

0.1

0.2

0.3

0.4

0.5

1/

z

Dynamical exponent

x2 (t), Bar Lev, 24

x2 (t), this work, 24

Gtyp0 (t), Luitz, 24

Gβ0 (t), Khait, 22

Gβ0 (t), Khait,∞

Gβ0 (t), Agarwal, 16

0 1 2 3 4 5

W

Exponent relations

x2 (t), this work, 24

σ (ω), Agarwal, 16

σ (ω), Khait,∞S(t), Luitz, 28

j(L), Znidaric, 400

j(L), Znidaric, 100

Figure 1. Left panel: Dynamical exponent 1/z. Here weshow the finite size results from the Supp. Mat. of Agarwalfor L = 16 et al. [33] and the ED result for L = 22 Khaitet al. [129], obtained from the decay of the infinite temper-ature correlation function Gβ=0

0 (t), as well as the result inthe thermodynamic limit using a variational extrapolation ofrecurrents (VER) by Khait et al. [129]. We also show ourown result obtained from the same quantity calculated for a(typical) pure state with definite energy, where the functionalform of the decay is fitted to include the oscillations [101].Full lines are the exponents as obtained from the width ofthe excitation x2(t) from Ref. [32] (infinite temperature) andcalculated for this work for the same typical pure state asmentioned before. Right panel: Dynamical exponent es-timated through the relations between the exponents fromEq. (33), compared to the best estimate from our calculationof x2(t) described in the left panel. The exponent extractedfrom the ac conductivity σ(ω) calculated for a finite systemsize (16) [33], we also show the thermodynamic limit resultobtained from σ(ω) exponent calculated by Khait et al. usingVER [129]. We include the best estimate of the exponent ofthe current scaling with system size from Žnidarič et al. [122]as well as a our analysis of same data restricted to L < 100,and the exponent from the entanglement growth power lawfrom Luitz et al. [142].

more dramatic, therefore rare region effects are overar-chingly called Griffiths effects [166]. It was proposed byAgarwal et al. that the subdiffusive ergodic phase, whichwas dubbed the Griffiths phase, could be effectively de-scribed by a one dimensional random chain governed bythe Master equation [33],

dPndt

= Wn,n−1 (Pn−1 − Pn) +Wn,n+1 (Pn+1 − Pn) ,

(34)where Pn is the probability to find a particle on site nand Wn,n+1 = Wn+1,n > 0 are the corresponding tran-sition rates, which are taken to be independent randomvariables. This equation had numerous appearances invarious contexts. It was first considered by Dyson, morethan half a century ago, who calculated the density ofstates of a random harmonic chain [167]. Replacing Pn

12

by an electric potential on a node n and Wn,n+1 by ran-dom conductances this model is equivalent to a randomresistor model, which was used in the hopping conductiv-ity literature [168, 169]. It was also used as a phenomeno-logical model to describe slow relaxation in spin glasses[160–163]. The properties of this model for various dis-tributions of Wn were extensively studied in the 80s byAlexander [136]. It was established that the most im-portant property of the distribution p (W ) is whether its⟨W−1

⟩moment exists. If this moment is finite [170] the

random chain is diffusive with, P0 ∼ t−1/2 and x2 (t) ∼ t.Otherwise, the system is subdiffusive with an anomalousdiffusion which depends on the details of the distribution[136]. For a power law distribution p (W ) ∼W−α, whichwas also the distribution considered in Ref. [33], it wasrigorously derived that the return probability asymptot-ically scales as [171],

P0 (t) ∼ t−(1−α)/(2−α) when Pn (t = 0) = δn0,(35)

with a Laplace transform, P0 (ω) ∼ ω−1/(2−α). To de-rive the generalized diffusion coefficient one assumes thescaling form [172],

Pn (ω) ≈ P0 (ω)F

(n

ξ (ω)

), ω → 0, (36)

where ξ (ω) is some correlation length and F (0) = 1.Due to the normalization

∑n Pn (ω) = ω−1, and(

ωP0 (ω))−1

≈∑n

F

(n

ξ (ω)

)≈ 2

∫ ∞0

dxF

(x

ξ (ω)

).

(37)

Changing the integration variables x′ = x/ξ (ω) gives,

ξ−1 (ω) ≈2ωP0 (ω)

∫ ∞0

dx′F (x′) . (38)

Now using the relation (14), one can write,

D (ω) =1

2ω2∑n

n2P0 (ω) =1

2ω2P0 (ω)

∑n

n2F

(n

ξ (ω)

).

(39)Changing the variables again and using (38) yields,

D (ω) =D0

ωP 20 (ω)

∼ ωα/(2−α), (40)

where D0 =∫∞

0dx′ x′2F (x′) /

[8(∫∞

0dx′F (x′)

)3] [172].Similar relationships between the exponents of the gen-eralized diffusion coefficient (which is proportional tothe ac conductivity) and the return probability wereobtained and verified numerically for the XXZ modelin Refs. [33, 137]. In dimensions higher than one therandom hopping model predicts asymptotically diffu-sive transport, since contrary to the situation in one

dimension, links with low transition rates (high barri-ers) can be avoided [173]. Nevertheless, transport ina two-dimensional disordered Anderson-Hubbard model(4) was studied by one of us in Ref. [117] and found to besubdiffusive for a broad range of parameters and withoutvisible crossover to diffusion at the studied times.

A simplified explanation of Griffiths effects was pre-sented by Gopalakrishnan et al. [137]. It assumed thatthe system is composed of a collection of independentlyrelaxing regions which additively contribute to the de-cay of the autocorrelation function, an approach familiarfrom the spin glass community [161]. The autocorrela-tion function is taken to be,

C (t) =⟨

e−t/τ⟩τ≡∫ ∞

0

dτ p (τ) exp [−t/τ ] , (41)

where p (τ) is the density of the regions with relax-ation time τ . Instead of using an exponential cutoff, inRef. [137] a sharp cutoff was assumed, namely,

C (t) =

∫ ∞t

dτ p (τ) . (42)

Moreover it was assumed that the density of regions andtheir corresponding relaxation rates are,

p (l) ∼ e−γld

τ (l) = eαl, (43)

where α and γ are constants, l is the linear dimension ofthe region and d is the dimension of the system. We notethat these assumptions are reasonable only for autocor-relation functions which do not decay to zero in the MBLphase, since only for these correlation functions the relax-ation time diverges at the transition. An existence of rarespatial regions created by rare local realizations of thedisordered potential is also implicitly assumed. Therefore(43) is not expected to hold for deterministic potentialssuch as the Aubry-André model [137]. From (43) one cancalculate the corresponding distribution function of therelaxation times is,

p (τ) ∼ 1

τexp

[− γ

αdlnd (τ)

]. (44)

For d = 1 the distribution of the relaxation times is givenby a power law and the integral in Eq. (42) can be eval-uated,

C (t) ∼ t−γ/α, (45)

which yields subdiffusive relaxation for γ < 2α, and asuperdiffusive relaxation otherwise. For d ≥ 2 approxi-mating the integral (42) by the largest integrand yields,

C (t) ≈ exp[− γ

αdlnd t

], (46)

which relaxes faster than any power law, yet slower thanexponential [137]. The procedure above is somewhat ar-bitrary since it strongly depends on the assumed distri-butions (43). While they have a clear physical meaning,

13

a more microscopic justification would be preferable. An-other problem with this approach is that it neglects thedependence between the different regions. While this is areasonable approximation in higher dimensions, for onedimensional systems it overestimates the relaxation ratesince neighboring rare regions should suppress the relax-ation of their surrounding. For example, naively calcu-lating the typical autocorrelation function,

C (t) ∼ exp [−t/ 〈τ〉] , (47)

yields exponential relaxation, since the average relax-ation time 〈τ〉 is finite even for one-dimensional systems(for α < γ < 2α). To correct for this discrepancy onehas to take into account the dependence between the re-gions, which was heuristically performed in Ref. [137].For a more detailed discussion on the Griffiths effects werefer the reader to Ref. [36].

B. Phenomenological Renormalization Group

Several real space phenomenological renormalizationgroup (RG) approaches were developed to study the uni-versal features of the MBL transition [105, 106, 174] (fora review see Ref. [35]). A real space coarse-graining ofthe system is performed, accompanied by a subdivisioninto ergodic and nonergodic regions. The main differ-ence between the approaches is the way in which theseregions are identified and combined during the RG steps.The simplified RG scheme presented in Ref. [174] startsfrom a random sequence of ergodic and nonergodic re-gions of different lengths according to some initial dis-tribution. The RG step consists then of identifying theshortest region and merging it with the two neighboringregions. The new region will be ergodic or nonergodicaccording to a majority rule of the three regions. Usingthese RG rules the critical distribution can be derivedas also the limiting distributions of the ergodic and non-ergodic phases. Interestingly, this RG procedure pointsto a fractal nature of the nonergodic inclusions in theergodic phase.

This procedure can be viewed as a maximally simpli-fied version of the more detailed RG approach proposedin Ref. [105]. In this work the regions were characterizedby their many-body level spacing ∆i and an entangle-ment rate Γi, which is inversely proportional to the timeentanglement spreads across the region. In addition, aset of two-region parameters ∆ij and Γij is kept, whichcorrespond to the parameters one would obtain if twoneighboring regions were merged. The RG step then con-sists of merging two regions with the fastest (combined)entanglement rate Γij , after which the coupling Γk;ij tothe neighboring region k is renormalized. If the couplingbetween the regions is effective, namely Γij ∆ij andΓjk ∆jk the rates are renormalized according to (a)Γ−1ij;k = Γ−1

ij + Γ−1jk − Γ−1

j , removing double counting of

the transversal time of region j. On the other hand, ifthe coupling is ineffective, the new entanglement rate isobtained from second order perturbation theory via (b)Γij;k = ΓijΓjk/Γj . For the case when only one of thelinks is effective there is some arbitrariness in the choiceof the rules. If the effective link is between two ergodicregions the rule (a) is used and when the effective cou-pling is between an ergodic and nonergodic region rule(b) is used. While this RG flow does not permit to di-rectly obtain the entanglement entropy between blocks, itwas estimated from the lifetime of product states 1/Γijand the number of accessible states at a given energy1/∆ij , capturing correctly the transition from a volumelaw scaling in the ergodic phase to an area law scaling inthe nonergodic phase and accompanied by a broad distri-bution of the entanglement entropy close to the criticalpoint. Transport properties were studied by consideringthe scaling of the typical transport time li/Γi with thelength of the region li. It was found that transport inthe ergodic phase in the vicinity of the critical point issubdiffusive with a dynamical exponent smaller then 1/2,while the entanglement growth is sublinear in time.

In Ref. [106] a similar real space RG method was pro-posed. Unlike the procedures discussed above, in thisapproach only resonant regions are combined, and thenonresonant regions are left intact. This removes the ar-bitrariness in RG rules when an ergodic and nonergodicregions have to be combined. Each region has a lengthli and a bandwidth Λi, and all the regions are coupledusing a coupling strength Γi,j which exponentially de-creases with the distance between the regions. After tworegions i and j of length li (lj) are merged, the couplingΓij , the bandwidth Λij = Λi + Λj + Γij and the levelspacing δij = Λij/(2

li+lj −1) are renormalized. The cou-pling to other regions Γij;k has to be updated too. Thisis the central step of the renormalization procedure andinvolves analyzing all possible processes coupling the re-gions i, j and k. It depends on the energy mismatch δEikof the individually merged regions i, k or j, k, which is de-fined as the minimal energy difference in the spectrum ofthe merged regions. If Γik δEik, then the renormalizedcoupling can be computed in second order perturbationtheory as Γij;k = ΓikΓij/δEik, otherwise all three regionsare strongly coupled and the coupling is given by the ad-dition of the two transport times Γ−1

ij;k = Γ−1ki +Γ−1

ij . Outof all possible processes the largest coupling is retained.Iterating this procedure until all resonant regions are ex-hausted generates the largest resonant “backbone” in thesystem. If this backbone percolates across the entire sys-tem the system will be ergodic, and nonergodic otherwise.

Potter et al. [106] identify subdiffusive transport inthe ergodic phase from a broad power law distributionp(τij) ∝ τ−αij of the transport time scales τij = 1/Γijwith a divergent mean (1 < α < 2). The authors arguethat transport can be viewed as a random walk (withbroadly distributed hopping rates) on the resonant back-

14

bone which yields a dynamical exponent of transport ofz = α/ (α− 1). In contrast to transport, entanglementis not a conserved quantity and spreads deterministicallyacross the chain, thus leading to a different dynamicalexponent zent = 1/ (α− 1) = z − 1.

We emphasize that the RG procedures described aboveare completely phenomenological and are not derivedfrom any microscopic model. A completely different realspace RG method, which is microscopically based hasbeen introduced in Refs. [133, 175], generalizing the ideaof strong disorder RG approaches for ground state prop-erties [176–179]. We refer the reader to the original worksin Refs. [133, 175, 180].

V. NUMERICAL METHODS

In this section we will describe some of the numeri-cally exact and approximate methods which can be usedto study the many-body problem. We note that thismethods are not limited to the prototype model we haveconsidered in Sec. II. Through this section we designatethe Hilbert space dimension by N and note that it scalesexponentially with the system size, L, e.g. for spin- 1

2systems it grows like N = 2L.

A. Exact methods for nonequilibrium timeevolution

1. Full diagonalization

Studying the properties of strongly correlated quan-tum systems is a formidable problem and an exact treat-ment of models is often possible only numerically. In atypical nonequilibrium numerical experiment the systemis prepared in some initial state |ψ0〉, which is not aneigenstate of the Hamiltonian matrix H ∈ CN×N . Thepropagation of the state in time can be performed by ex-actly diagonalizing the Hamiltonian H = UDU†, wherethe matrix D = diag (E0, . . . , EN−1) is diagonal and con-tains the eigenvalues En of H while the columns of Ucorrespond to the orthonormal eigenvectors |n〉 (cf. Sec.VB for more details). The solution of the Schrödingerequation for the time dependent wave function is given by|ψ(t)〉 =

∑n e−iEnt |n〉 〈n|ψ0〉, where 〈n|ψ0〉 are the coef-

ficients of the initial wave function in the eigenbasis of theHamiltonian. If the initial wavefunction is represented asa vector x0 ∈ CN in the computational basis, then thewavefunction at time t is obtained by x(t) = U†e−iDtUx0,where the matrix exponential of the diagonal matrix Dis trivial. While this method is able to access arbitrarilylong times, it is limited by the exponential growth of theHilbert space with the size of the system. The computa-tional complexity of this method is about O

(N 3), and

the required memory is O(N 2), effectively limiting the

applicability of the method to lattice sizes of . 16 (if thesystem has no additional symmetries).

2. Krylov space time evolution

Nautts and Wyatt realized in 1983 [181] that one canavoid the full diagonalization of the Hamiltonian by us-ing a Krylov space method to calculate the exact timeevolution |ψ (t+ ∆t)〉 = e−iH∆t |ψ (t)〉. Using the seriesexpansion of the exponential, we obtain

e−iH∆t |ψ (t)〉 =

∞∑k=0

(−i∆t)k

k!Hk |ψ (t)〉 , (48)

which for very small ∆t may be used directly, butis numerically inherently unstable [182]. To obtain amore stable expansion, it is useful to note that thewave function at time t + ∆t is well approximated bya vector in the m dimensional Krylov space Km =

span(|ψ (t)〉 , H |ψ (t)〉 , H2 |ψ (t)〉 , . . . , Hm−1 |ψ (t)〉

).

Based on this observation, an orthonormal basis ofthe Krylov space Km is iteratively generated usingthe numerically stable Arnoldi algorithm [183] and theHamiltonian is projected into this subspace after miterations, yielding [182]

e−iH∆t |ψ (t)〉 ≈ Vme−iV†mHVm∆te1. (49)

Here the columns of the matrix Vm ∈ CN×m contain theorthonormal basis vectors of the Krylov space Km, ande1 ∈ Cm is the first unit vector (which corresponds to|ψ (t)〉 in the new basis as this is the first column of Vm).Note that the matrix V†mHVm ∈ Cm×m is an upper Hes-senberg matrix of small dimension m N , which canbe readily exponentiated using standard methods, suchas a Padé approximation or a rotation to the eigenbasis.The dimension of the Krylov space m is continuouslyincreased until the wavefunction is converged to the de-sired precision. This method is very powerful since it ex-ploits the sparseness of the Hamiltonian and does not re-quire its full diagonalization. The memory requirementsand the computational complexity of this approach aremuch more favorable compared to exact diagonalization(see Table I). This approach has been used to study thenonequilibrium dynamics of spin chains with lengths upto L = 28 [51, 142, 154].

3. tDMRG

An independent approach to obtain the numericallyexact time evolution of the wave function after a quenchemploys a representation of the wave function as a matrixproduct state (MPS). For models for which the Hamilto-nian can be decomposed into terms which operate on two

15

adjacent sites, which we will call bond terms, the prop-agation of the wavefunction in time is quite straightfor-ward. In order to calculate the wavefunction after a timestep ∆t, the Hamiltonian is decomposed into two termsH = Heven + Hodd, where Heven and Hodd contain even(odd) bond terms. While Heven and Hodd need not com-mute, all terms within Heven

(Hodd

)commute with each

other. Therefore a Trotter decomposition of the time evo-lution operator [cf. Eq. (48)] can be used to time evolvethe state by ∆t. The simplest decomposition leads to anerror of ∆t2,

e−iH∆t = e−iHeven∆te−iHodd∆t +O(∆t2

), (50)

however, higher order decompositions can be used (cf.Refs. [184, 185]. Note that as the matrix exponentials onthe right hand side of Eq. (50) contain only commutingterms, they can be applied sequentially in one DMRGsweep. During the application of the odd and even bondterms to the MPS, the bond dimension of the MPS isadaptively truncated such that the discarded weight, i.e.the sum of the discarded singular values does not exceeda certain threshold. As the number of retained singularvalues directly limits the maximal entanglement entropythat can be encoded by the MPS, it is clear that thebond dimension of the MPS has to grow exponentiallywith the entanglement entropy. In the ergodic phase theentanglement entropy after a quench from a product stategrows as a power law in time [142], thus leading to astretched exponential growth of the bond dimension withtime and effectively limiting this method to short times.In the MBL phase the situation is more favorable sincethe entanglement entropy grows logarithmically in time[186–189], leading to only a power law growth of the bonddimension. For details on the method, we refer the readerto the original papers on this adaptive method by Vidal[190, 191] and to a review on DMRG [192].

4. tDMRG for open systems

The study of transport properties can be convenientlyperformed by opening the system and attaching it to two(or more) leads with a different chemical potential. ForMarkovian leads and under additional approximationsthe evolution of the density matrix of the system ρ canbe described using the Lindblad equation [193],

d

dtρ ≡ Lρ = i

[ρ, H

]+ γ

∑k

([Lkρ, L

†k

]+[Lk, ρL

†k

]),

(51)where the Lindblad operators Lk, describe the couplingbetween the system and the bath. The Lindlad equa-tion can be numerically solved using tDMRG [194, 195].The evolution of the density matrix is performed by in-creasing the size of the Hilbert space and considering

the density matrix operator ρ(t) as a vector in the en-larged space, whose time evolution is governed by theLiouvillian L (cf. (51). In this enlarged Hilbert spacethe tDMRG method described in the previous sectioncan be applied and it appears that in many cases theentanglement entropy in the operator space, which gov-erns the bond dimension and therefore the efficiency ofthe method, grows slowly in time due to decoherenceeffects caused by the Markovian bath. This favorablecomputational complexity allows to reach the nonequilib-rium steady state (NESS) at long times, and to calculatethe stationary magnetization and the stationary current[122]. If the bias between left and right leads is smallenough, the system is in the linear response regime andthe current in the NESS reveals the nature of the trans-port. Žnidarič et al. have used this method to study thedynamical exponent in the random XXZ chain for sys-tem sizes up to L = 400, arguing in favor of a transitionbetween a diffusive and a subdiffusive regime at weakdisorder strength [122]. For strong disorder this methodbecomes increasingly expensive since the time it takes toreach the stationary state increases.

5. Dynamical typicality

Quantum typicality can be viewed as a geometri-cal concept that follows from Lévy’s Lemma. Thislemma states that for a Lipschitz-continuous functionf : S(2n−1) → R defined on the surface of a high dimen-sional sphere, any point x ∈ S(2n−1) drawn randomlyfrom a uniform distribution on the sphere will yield f(x)exponentially close to the average of f over the surfaceof the sphere [196].

Since any normalized quantum state in a finite dimen-sional Hilbert space of dimension N can be representedas a point on the surface of a 2N dimensional unit hy-persphere S(2N−1), and the trace of an operator Tr O canbe written as the integral of O over the surface of thissphere with respect to the Haar measure, it follows thatthe expectation value of O for any random pure state |ψ〉on the sphere is exponentially close to the value of thetrace, if the operator can be represented as a Lipshitz-continuous function on the hypersphere. This is typicallythe case for local operators. More precisely, the proba-bility to deviate from the trace by more than ε > 0 isexponentially small,

P[∣∣∣Tr O − 〈ψ| O |ψ〉

∣∣∣ ≥ ε] ≤ ae−bN ε2

, (52)

with positive constants a and b. This means that thetrace of the operator O can be replaced by an expectationvalue obtained from a random pure state |ψ〉 to a preci-sion which improves for larger Hilbert space dimension[197–203]. To illustrate its application, we demonstratehow it can be used to calculate a correlation function inthe canonical ensemble:

16

CβO(t) =1

ZTr(

e−βHO (t) O)≈ 1

〈β|β〉

⟨β∣∣∣O (t) O

∣∣∣β⟩ ,(53)

where β is the inverse temperature. Here we have usedthe cyclic property of the trace, applied Lévy’s lemmasubstituting the trace by an expectation value of a ran-dom state |ψ〉 and finally defined |β〉 ≡ e−

β2 Ht |ψ〉 (cf.

[203]). This state can be efficiently calculated by imag-inary time evolution of the pure state |ψ〉, followedby real time evolution to obtain the correlation func-tion. This task can be performed either by integra-tion of the Schrödinger equation using the recently de-veloped Runge-Kutta schemes [204, 205], or by utilizingthe Krylov space technique discussed in the previous sec-tion. All these approaches can be applied without fulldiagonalization of the Hamiltonian and rely solely on theability to calculate the matrix vector product Hx, whichcan be achieved even without storing the sparse Hamilto-nian matrix H. The memory requirement is thus reducedto the size of a few Hilbert space vectors. We remarkthat in Ref. [101], we have applied a simplified version ofthis approach by creating a microcanonical typical state,which we called “energy squeezed state”. This state wasconstructed by applying powers of (H −σ)2 to a randomvector in the Hilbert space to suppress contributions fromeigenstates far away from the target energy σ.

B. Exact methods for eigenstates calculation

The absence of transport is the defining property thatdistinguishes the MBL phase from the ergodic phase.Transport can be efficiently studied using the numeri-cal methods described in the previous section. However,the MBL transition can also be viewed as an eigenstatephase transition (cf. Ref. [35]), which is characterizedby strikingly different properties of the eigenstates of theHamiltonian in the ergodic and nonergodic phases, butalso by different statistical properties of the energy spec-trum. To study this aspect numerically, it is thereforeimportant to be able to calculate some or all or the eigen-values and eigenstates of the Hamiltonian.

1. Full diagonalization

Clearly, the first choice to obtain exact high energyeigenstates [206] is the full diagonalization of the Hamil-tonian. This is typically done using the standard protocolfor dense matrices: First, the Hamiltonian is brought totridiagonal form by Householder reflections, the tridiag-onal matrix is then diagonalized by efficient algorithms,such as the divide and conquer [207] approach or us-ing multiple relatively robust representations [208], and

finally the obtained eigenvectors are transformed backto the original basis using the Householder transforma-tions of the first step in inverse order. This recipe isavailable in highly optimized LAPACK implementations formany architectures, yielding high precision results. Sincethese methods are based on dense matrices, they requireO(N 2) memory to store the dense matrix (in addition toO(N 2) work space for the divide and conquer algorithm)as well as to store all eigenvectors of the result. The com-putational complexity is dominated by the Householderstep and it scales as O(N 3).

2. Subset diagonalization

For some applications only a few eigenstates and eigen-values within some interval [E−, E+] are required. Theshift-invert method is the current state-of-the-art methodto tackle this problem. It is closely related to inverse iter-ation and relies on the fact that the extremal eigenvaluesof (H−σ)−1 correspond to the eigenvalues of H which areclosest to the target energy σ. Furthermore, while thetypical scaling of the level spacing of the original prob-lem is N−1, the level spacing in the corresponding partof the transformed spectrum is N . Therefore standardKrylov space methods, such as the Lanczos algorithmor the Arnoldi iteration can be efficiently used to obtainseveral of the highest and lowest lying eigenvalues andeigenstates of the transformed problem. The eigenval-ues of the transformed problem are trivially transformedback to the original problem and the eigenvectors areinvariant under this transformation and therefore are di-rectly obtained.

The hardest part of this procedure is the repeated cal-culation of the action of (H − σ)−1 on vectors duringthe application of Krylov space methods. This is typ-ically done by first decomposing the Hamiltonian intoupper and lower triangular matrices (the LU decomposi-tion) such that (H−σ) = LU using Gaussian elimination.Subsequently LUx = b is solved, yielding x = (H−σ)−1b.For the calculation of the LU decomposition, efficient im-plementations that exploit the sparseness of H are avail-able. For example, for distributed memory machines theMUMPS [209, 210] and SuperLU [211] libraries can beused. The shift-invert technique has been used to mapthe energy-disorder phase-diagram of the XXZ model (1)by one of us [41].

3. Excited state DMRG

While matrix product state methods are extremelysuccessful for the study of ground-state properties of onedimensional systems, they were typically not employedto find matrix product state (MPS) representations ofhighly excited eigenstates of the Hamiltonian, which only

17

recently became of interest in the context of MBL. Dueto the area law entanglement of eigenstates in the MBLphase it is natural to expect that such a representa-tion will be efficient. In the ergodic phase, on the otherhand it is highly inefficient due to the volume law scal-ing of entanglement. Recently, several groups have de-veloped methods to find highly excited eigenstates withMPS based methods, which work well in the MBL phase.Yu et al. developed SIMPS (shift invert matrix productstate method) that relies on the idea of the shift-invertmethod [212], trying to find an MPS which best approx-imates the eigenstate of (H−σ)−1 with the largest mag-nitude eigenvalue. For this purpose, Yu et al. propose a

method, which is used to iteratively apply(H − σ

)−1

toan initial MPS |ψ0〉. This iteration is converging expo-nentially fast (at sufficiently large bond dimension) to aneigenstate of (H − σ)−1 which corresponds to an excitedstate of the Hamiltonian with an eigenvalue close to σ.The key insight of the method is that the next iteration|ψk〉 could be thought of as a solution of a variational

minimization problem∥∥∥(H − σ) |ψk〉 − |ψk−1〉

∥∥∥2

where|ψk−1〉 corresponds to the previous iteration. The solu-tion of this problem is obtained using an adapted versionof the DMRG sweeping protocol. This method was subse-quently used by Serbyn et al. to study the entanglementspectrum in the MBL phase [213].

As SIMPS relies on a modification of DMRG, a simplermethod was proposed to obtain MPS representations ofhighly excited eigenstates in DMRG [212, 214]. Instead oftrying to invert the global Hamiltonian, it is based on thelocal effective Hamiltonians appearing during the DMRGsweep. The local matrices of the MPS are updated bychoosing an excited eigenstate of the local Hamiltonianand yielding an eigenstate of the global Hamiltonian notnecessarily close to a target energy. The procedures ofselecting the eigenstates of the effective Hamiltonian ei-ther relies on choosing the eigenstate with energy closestto the energy of the previous MPS [212], or on the prop-erty of the MBL phase that the eigenstates are very closeto product states, such that an eigenstate which has themaximal overlap with the MPS of the previous iterationis selected [214]. This approach circumvents the generalproblem that for large system sizes, the energy level spac-ing of the full spectrum becomes smaller than machineprecision. Other approaches exploit the idea of “spectumfolding”, noting that the groundstate of (H − σ)2 corre-sponds to the eigenstate of H with an eigenvalue closestto σ [215, 216]. All these methods are currently employedonly in the MBL phase and it is unclear whether they willbe useful to study the physics very close to the transitionor in the ergodic phase due to the presence of high entan-glement entropy. We note that in the fully MBL phaseit has been argued that the complete spectrum can beencoded in a single matrix product operator [217–220].

Time evolution memory CPU L time

ED O(N 2) O(N 3) ≈ 18 ∞Krylov O(mN ) O(LNtN ) ≈ 30 tmax

tDMRG O(Lχ2) O(LNtχ3) > 100 ≈ O(lnχ)

Table I. Comparison of numerical methods for time evolution.Here m is the number of Krylov vectors, Nt is the number oftime steps, χ is the bond dimension, N is the Hilbert spacedimension and L is the system size.

Eigenstates memory CPU L comment

ED O(N 2) O(N 3) ≈ 18 shared memory

Shift-Invert O(N 2) O(N 3) ≈ 22 distributed memory

ES-DMRG O(Lχ2) O(Lχ3) ≈ 100 MBL only

SI-DMRG O(Lχ2) O(Lχ4) ≈ 100 MBL only

QMC O(L) O(

1σ2MC

)≈ 100 MBL only, approx.

Table II. Comparison of numerical methods for the extrac-tion of high energy eigenstates. The notation is the same asin Table I, σ2

MC is the target variance of QMC result. TheQMC method mentioned here is currently not a strictly exactmethod [221].

4. Quantum Monte-Carlo

Quantum Monte-Carlo (QMC) methods are extremelyuseful to study equilibrium finite temperature physics aswell as low temperature properties provided that there isno sign problem. However, they are not able to resolvesingle eigenstates which are not groundstates. Inglis andPollet have recently made progress in this direction by ef-fectively shifting the energies of the original Hamiltonianto make a highly excited eigenstate the new groundstate[221]. This is achieved by exploiting the fact that eigen-states of MBL systems can be labeled by eigenvalues ofon extensive number of conserved quasilocal quantities[222, 223]. This method is conceptually new and verypromising, although its current implementation relies onan approximate construction of the quasilocal conservedoperators with constraints on their analytic form to makethem compatible with the worm algorithm. By construc-tion, this method is only useful to study the MBL phase.

C. Approximate method: Perturbation theory

All numerically exact methods have strong size or timeconstraints, which result in finite size effects, especiallyin the limit of weak disorder. These constraints areeven more pronounced in higher dimensions, where cur-rently there are no efficient methods for an exact studyof nonequilibrium dynamics. Access to larger systemsand longer times can be gained by utilizing approximatemethods. Below we survey one such approach which wassuccessfully applied for the study of transport in one-

18

dimensional [31] and a two dimensional system by one ofus [117].

For the MBL problem this method was introducedin the work of Basko, Aleiner and Altshuler [18]. Themethod is perturbative in the interaction strength and aswas demonstrated by one of us, is able to quantitativelyreproduce numerically exact results in the limit of largedisorder. For very weak disorder, the method becomesincreasingly unreliable and tends to overestimate the re-laxation in the system [31, 117]. We note that whileperturbation theory is clearly an analytical tool, its nu-merical implementation requires the solution of certainnumerical difficulties (for details see [224]). The quanti-ties of interest are one particle correlation functions,

G>ij (t; t′) = −iTr(ρ0ci (t) c†j (t′)

)(54)

G<ij (t; t′) = iTr(ρ0c†j (t′) ci (t)

),

where ρ0 is the initial density matrix and c†j createsa spinless fermion at site j. For a noninteractinginitial density matrix, the Green’s functions obey theKadanoff–Baym equations of motion [225],

i∂tG≷ (t, t′) =

(h0 + ΣHF (t)

)G≷ (t, t′)

+

∫ t

0

ΣR (t, t2)G≷ (t2, t′) dt2

+

∫ t′

0

Σ≷ (t, t2)GA (t2, t′) dt2, (55)

where spatial indices and summations are suppressed forclarity, h0,nm is the one particle Hamiltonian, ΣHF (t),Σ≷ (t) are the Hartree-Fock, greater and lesser self-energies of the problem respectively; and the superscripts’R’ and ’A’ represent retarded and advanced Green’sfunctions and self-energies, which are defined as

ΣR (t, t2) = θ (t− t2)(Σ> (t, t2)− Σ< (t, t2)

)(56)

GA (t2, t′) = −θ (t′ − t2)

(G> (t2, t

′)−G< (t2, t′)).

Since the exact form of the self-energies is normally un-known, they are commonly approximated up to someorder in the small parameter of the problem. For theproblem which is the subject of this review the naturalsmall parameter is λ ≡ U/δ where U is the interactionstrength and δ is the typical energy difference of nearbylocalized single-particle states δ ≡ ∆/ξ. Here ∆ is thesingle-particle bandwidth and ξ is the single-particle lo-calization length. To second order in λ a particularlyuseful approximation is the self-consistent second-Bornapproximation,

ΣHFij (t) = −iδij∑k

VikG<kk (t; t) + iVijG

<ij (t; t)

Σ>ij (t, t′) =∑k,l

VilVjkG<kl (t

′, t)× (57)

[G>lk (t, t′)G>ij (t, t′)−G>lj (t, t′)G>ik (t, t′)

],

where Vij = V (δi,j+1 + δi,j−1) is the interaction poten-tial. Using this approximation one can write a closedform equation for the correlation functions (54), which isthen solved numerically. This method requires a memorywhich scales like O

(L2N2

t

), where Nt is the number of

time steps required to solve (55) to predetermined pre-cision. The computational complexity of this methodscales like O

(L3N3

t

), and could be further reduced to

O(L3N2

t

)by making additional approximations [226–

228]. For more technical details on this method as alsoto detailed comparison to exact methods the the readeris referred to Refs. [31, 117, 228].

VI. DISCUSSION AND OPEN QUESTIONS

In the previous sections we reviewed in detail the cur-rent knowledge of the ergodic phase at weak disorder,preceding the MBL transition. Here, we will identifysome important open questions and discuss the progressthat has been made towards answering them. In Fig. 2we present a visual summary of the results as also someof the open questions. It is apparent, that while recentworks identified fascinating possible scenarios for the richphysics of the ergodic phase, the overall picture is not yetsettled. Future works in this field will have to clarify howthe observed phenomenology evolves as a function of sys-tem size to put existing contradictions into perspective.

W1 2 3 40

MBL

Subdiffusive

Energy trans. Diffusive? Subdiffusive

Critical? Poisson?IntermediateWD ?

Multifractality

Valid in a generalized form

?

(a)

(b)

(c)

(d)

(e)

Eigenvalue stat.

ETH

Spin/EE trans.

?

Figure 2. A visual summary of current results and openquestions on the ergodic phase of the XXZ model (1) withJz = 1. Different color patches represent different phases assuggested by various studies. The locations of transitions orcrossovers between the different phases are presented only ap-proximately and are displayed as sharp for better readability.Question marks represent open questions and arrows indicatethat some studies suggest that the phase shrinks to the criticalpoint Wc ≈ 3.7 in the thermodynamic limit. (a) The valid-ity of ETH was studied in Refs. [101, 142], (b) the general-ized fractal dimensions D1,2 were studied in Refs. [41, 92, 93],(c) a detailed study of the eigenvalue statistics was done inRefs. [71, 93], (d) energy transport was studied in Ref. [154],(e) spin transport and entanglement spreading was studiedin Refs. [32, 33, 129, 142, 152].

19

A. Subdiffusion and the subdiffusion to diffusiontransition

While the MBL phase can be defined by an absence oftransport, the nature of transport in the ergodic phaseis not a priori clear. Many numerical studies have ad-dressed this question after first evidence for subdiffusivetransport in a one-dimensional XXZ model was found[32, 33]. The results of most numerical studies are con-sistent with the interpretation that at intermediate dis-order 1 .W . 3.7 spin transport is subdiffusive and theentanglement growth is sublinear [32, 33, 129, 142, 152],with a continuously varying dynamical exponent z, whichdiverges at the MBL transition. In Ref. [154] it was ar-gued that for 1 . W . 2.5 energy transport is diffusivewhile spin transport is subdiffusive. This study howeveris in contradiction with Ref. [142], since if true asymp-totically in time, it would suggest that energy was trans-ported faster than information (entanglement entropy).For very weak disorder, W < 1, some studies yet findsubdiffusive spin transport [142], while others argue infavor of a transition to diffusion [33, 122]. Currently,the most compelling evidence stems from the study ofan open XXZ chain with system sizes up to L = 400by Žnidarič et al. [122]. This work argues in favor of atransition between diffusive and subdiffusive behavior atW ≈ 0.6. While this work cannot rule out weak subdif-fusive transport for W < 0.6, which might occur for evenlarger system sizes (see right panel of Fig. 1, and dis-cussion at the end of Sec. III C 6), it points out that EDstudies in this region of parameters are subject to severefinite size effects. Interestingly, in this region, W < 0.6the fluctuations of local operators in the eigenbasis of theHamiltonian are perfectly Gaussian, verifying exactly theETH ansatz [100, 101]. However currently no direct con-nection between the nature of transport and the shape ofthe probability distributions is known and we can onlyspeculate that such perfectly Gaussian distributions are asign for diffusion, while heavily tailed non-Gaussian dis-tributions may signal subdiffusion. The situation in di-mensions higher than one is less clear, since numericallyaddressing transport for d ≥ 2 remains very challeng-ing. Algorithmic progress and input from experiments isrequired to clarify the nature of transport in this case.Currently there is only one numerical study which pointstowards subdiffusion in two dimensions based on pertur-bation theory (cf. Sec. VC) [117].

This seemingly clear picture of transport in one-dimensional systems is disturbed if the numerical evi-dence is quantitatively compared. We have presented acomparison of recent numerical estimates of the dynam-ical exponent obtained by various methods. By lookingon Fig. 1 it is obvious that the results match only qualita-tively, moreover some commonly used relations betweenthe exponents [cf. Eq. (33)] do not hold. While thismight indicate that some of these relations should be

reconsidered, the observed disagreement between the ex-ponents could also follow from the difficulty of extractingdynamical exponents from numerical calculations on fi-nite systems. In fact, a very recent work evaluated thespread of spin perturbations starting from initial con-ditions with fixed energy density [229]. In this work aconvergent (with system size) dynamical exponent couldnot be obtained and it was argued that the observed sub-diffusion is a transient [229]. Asymptotic diffusion andfinite dc conductivity in the whole ergodic phase werealso found in ac conductivity studies [124, 125], usingsystem sizes up to L = 28 similar to the studies whichobserve subdiffusion. Future work will have to resolvethis contradiction.

B. Existence of the “bad metal”

The observation of intermediate level statistics as wellas of multifractal distributions of local operators at dis-order strengths W & 2 [71, 92] is consistent with theprediction of the existence of a delocalized, nonergodicphase (dubbed “bad metal” by Altshuler [76]) for disor-der strengths below the MBL transition [75]. However,the question whether this phase shrinks in the thermo-dynamic limit to a critical point or remains of finite ex-tent in the parameter space is still open [92, 93]. Largescale studies on random regular graphs (RRGs) seem topoint out that this phase disappears in the thermody-namic limit [87–89], although there is also no consensushere [82–90]. Moreover while both problems are related,it is not clear a priori that results from RRGs apply forphysical models.

The multifractal nature of this phase suggests thatit might be related to the observed subdiffusion, andcould also explain the poor agreement between the var-ious dynamical exponents, as presented in the summaryof Sec. III C (see Fig. 1). While a direct relation betweenmultifractality in the many-body space and real spacesubdiffusion was not established, a step in this directionwas performed in Refs. [71, 92, 101]. There are howevera few problems with this interpretation: (i) subdiffusionappears to persist in the thermodynamic limit [122] whilethe “bad metal” seems to shrink in this limit [92], (ii) look-ing at Fig. 2 it is clear that the extent of observed sub-diffusion is much larger then the extent of the observedmultifractality (iii) in this region the definitions of ergod-icity via ETH and ergodicity (via eigenvector statistics)do not agree, suggesting that the definitions of ergodicityas discussed in Sec. III A are not equivalent.

Another interesting question is whether there is a re-lation between the “bad metal” and the Griffiths effectspicture presented in Sec. IV.

20

C. Mechanism of subdiffusive transport

Even though the numerical evidence is not univocal,the existence of a subdiffusive regime is certainly a validscenario consistent with many numerical studies. Theproposed mechanism for subdiffusive transport, dubbedGriffiths effects, is the existence of rare insulating regionswhich serve as bottlenecks for the transport of particlesand entanglement [33, 137]. While this mechanism is dif-ficult to test numerically, it seems consistent with spatialvariations in the entanglement structure as observed nu-merically [100, 107, 110]. However, other indirect testsof the predictions of the Griffiths picture reveal severalissues: (i) it predicts asymptotic diffusion in dimensionshigher then one (cf. the review [36]), which seems to con-tradict the observation of subdiffusion in two dimensions[117] (ii) it predicts diffusion for quasiperiodic potentials,although sublinear entanglement growth was observed forthe Aubry-André model in Ref. [230] as well as subdif-fusive transport in Refs. [231, 232] (iii) Griffiths effectsare expected to be subdominant far from the MBL tran-sition, which is in contrast to the extended subdiffusivephase found in most studies. Due to the above, we feelthat Griffiths picture may have to be refined in futureworks. In particular its relation to observed signaturesof multifractality has to be better understood.

D. Relation to classical disordered models

An important point that we did not discuss in this re-view is the connection between the quantum and classicaldisordered models. The pertinent question in the contextof the ergodic phase is to which extent the ergodic phase,which is the subject of this review, can be considered clas-sical. A pioneering study in this direction suggests thatclassical disordered models are diffusive [67]. We referthe reader on a recent review on this subject [233].

The MBL transition is in many ways similar to theglass transition [18]. In particular commonly used MBLmodels, such as (1) and (4), are superficially reminiscentof spin-glass models. Both have quenched disorder and afinite temperature ergodic–nonergodic transition. How-ever while much of the phenomenology is similar, thereare important differences. The spin glass transition is athermodynamic phase transition which occurs in a pres-ence of an external heat bath [234], while the MBL transi-tion does not appear to have a thermodynamic signatureand occurs only for isolated systems [18]. One of themost interesting questions in this context, is whether theergodicity breaking mechanisms of spin glasses and MBLare somehow related.

The relation to structural glasses is more remote, dueto absence of quenched disorder in the structural glassesmodels. Attempts to find a stable MBL phase for dis-order free, translationally invariant models were unsuc-

cessful [91, 235–245]. Notwithstanding, the ergodic phasewhich is the subject of this review, shares many proper-ties with supercooled liquids [115]. To point out theseparallels, we have used notation borrowed from the struc-tural glasses community in our discussion of the dynam-ical properties in Sec, III C. Similarly to the supercooledliquids the ergodic phase thermalizes, and the relaxationof density (spin) autocorrelation functions is subexponen-tial with a relaxation time which diverges at the transi-tion. Interestingly, the theory of the MBL transition asestablished in Ref. [18] is also reminiscent of the mode-coupling theory, which works remarkably well for struc-tural glasses [246]. Deeper connections between the twofields should definitely be explored in future works.

E. Summary

In this review we surveyed the features of the ergodicphase, which occurs in generic interacting systems withsufficiently weak quenched disorder. We have explainedin which sense this phase could be considered ergodic,and elaborated on the different notions of ergodicity inthis context. We presented the peculiar static and dy-namical properties of this phase, which is characterizedby intermediate eigenvalue statistics, signatures of mul-tifractality and the emergence of power-laws for almostany dynamical property, including the growth of the en-tanglement entropy. We have explained the phenomeno-logical rare-region picture (the Griffiths picture), and itspredictions on the relations between the different power-law exponents, as also the numerical verification of thesepredictions. Finally, we presented all available numeri-cally exact and approximate methods for the explorationof this phase and finished the review with discussion ofsome of the pertinent open questions.

We are grateful to Ilia Khait and Marko Žnidarič forsharing their original data. YB acknowledges fundingfrom the Simons Foundation (#454951, David R. Re-ichman). DJL was supported by the Gordon and BettyMoore Foundation’s EPiQS Initiative through Grant No.GBMF4305 at the University of Illinois. This researchis part of the Blue Waters sustained-petascale comput-ing project, which is supported by the National ScienceFoundation (awards OCI-0725070 and ACI-1238993) andthe state of Illinois. Blue Waters is a joint effort of theUniversity of Illinois at Urbana-Champaign and its Na-tional Center for Supercomputing Applications.

21

Appendix A: Relation between the mean-squaredisplacement and the current-current correlation

functions

For convenience of the reader we derive a general re-lation between the correlations of a conserved quantityand the correlations of the corresponding current, whichcould be useful to applications well beyond the scope ofthis review. Similar relations were derived previously, seeRefs. [247–249].

We focus on Hamiltonians which conserve the quantity,Q =

∑Lm=1 nm, namely

[H, Q

]= 0 , and for which the

following continuity equation applies,

∂nk∂t

= ∆jk, (A1)

where ∆jk ≡ jk − jk−1 is the backward discrete deriva-tive. We define the means square displacement (MSD) ofthe excitation in this density to be

x2 (t) =1

L

L∑k=1

L∑l=1

(k − l)2 Re 〈δnk (t) δnl〉 , (A2)

where 〈.〉 is the equilibrium average and δnk (t) ≡ nk (t)−〈nk〉. We note the following identity,

〈(nk (t)− nk) (nl (t)− nl)〉 = 2 〈δnkδnl〉 (A3)− 〈δnk (t) δnl + δnkδnl (t)〉 ,

which is true for expectation with respect to the equilib-rium state. To calculate MSD we multiply by (k − l)2

and sum twice over the lattice, which gives,

x2 (t)−x2 (0) = − 1

2L

L∑k,l=1

(k − l)2 〈(nk (t)− nk) (nl (t)− nl)〉 .

(A4)Using the continuity equation we can write,

nk (t)− nk =

∫ t

0

dt∆jk (t) , (A5)

such that

x2 (t)− x2 (0) = − 1

2L

∫ t

0

dt1

∫ t

0

dt2× (A6)

×L∑

k,l=1

(k − l)2⟨

∆jk (t1) ∆jl (t2)⟩.

Taking a partial sum twice and assuming periodic bound-ary conditions (or alternatively neglecting the boundaryterms) gives,

x2 (t)− x2 (0) =1

L

∫ t

0

dt1

∫ t

0

dt2

⟨J (t1) J (t2)

⟩, (A7)

where J (t) =∑Lk=1 jk (t) is the total current. Since any

correlation function in equilibrium depends only on the

time difference we change the variables to τ = t1−t2, andt1 = t1 which has an unity Jacobian and the followingtransformation of the integration boundaries,

x2 (t)− x2 (0) =2

L

∫ t

0

dt1

∫ t1

0

dτ⟨J (τ) J (0)

⟩, (A8)

which could also be written as,⟨J (t) J (0)

⟩=

d2

dt2x2 (t) , (A9)

which is very similar to its classical form. Taking theFourier transform we get the relation between the cor-responding frequency dependent diffusion coefficient andthe MSD (Eq. (14)),

D (ω) = −ω2

∫ ∞−∞

dt x2 (t) eiωt. (A10)

We note that this derivation assumes only the continuityequation (A1), periodic boundary conditions and an ex-pectation value with respect to the thermal state. It doesnot assume linear response and any knowledge aboutthe Hamiltonian except of the existence of the conservedquantity. It also does not assume any specific form of theconserved quantity or the corresponding current.

∗ dluitz@illinois.edu† yb2296@columbia.edu

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