Marginal Anderson localization and many body delocalization

Rahul Nandkishore1 and Andrew C. Potter2

1Princeton Center for Theoretical Science,Princeton University, Princeton, New Jersey 08544, USA and2Department of Physics, University of California, Berkeley, CA 94720, USA

(Dated: June 5, 2014)

We consider d dimensional systems which are localized in the absence of interactions, but whosesingle particle (SP) localization length diverges near a discrete set of (single-particle) energies, withcritical exponent ν. This class includes disordered systems with intrinsic- or symmetry-protected-topological bands, such as disordered integer quantum Hall insulators. In the absence of interactions,such marginally localized systems exhibit anomalous properties intermediate between localized andextended including: vanishing DC conductivity but sub-diffusive dynamics, and fractal entanglement(an entanglement entropy with a scaling intermediate between area and volume law). We investigatethe stability of marginal localization in the presence of interactions, and argue that arbitrarily weakshort range interactions trigger delocalization for partially filled bands at non-zero energy densityif ν ≥ 1/d. We use the Harris/Chayes bound ν ≥ 2/d, to conclude that marginal localizationis generically unstable in the presence of interactions. Our results suggest the impossibility ofstabilizing quantized Hall conductance at non-zero energy density.

I. INTRODUCTION

Edge states of topological systems give rise to a strik-ing set of coherent quantum phenomena. For example,chiral edge states of two-dimensional quantum Hall sys-tems lead to precisely quantized Hall conductance at zerotemperature. However in thermal systems with non-zerotemperature, these edge-state properties are washed outdue to thermal excitations that communicate betweenopposite edges of the system. Such deleterious thermaleffects can sometimes be mitigated by careful cooling andisolation. Nevertheless, it is interesting and potentiallyuseful to ask whether one can obtain systems with topo-logical edge state properties that are immune to thermalwashing out, circumventing the need for cooling.

An intriguing possibility arises from the study of many-body localization (MBL) - a phenomenon whereby well-isolated quantum systems fail to thermalize due to thelocalization of excitations by strong disorder [1–11]. InMBL systems, disorder can protect quantum coherenceagainst such thermal degradation by localizing the of-fending excitations. In this fashion certain coherentquantum phenomena including symmetry breaking andtopological order may survive in MBL systems at non-zero energy density (the appropriate analog of finite tem-perature in non-thermal systems) [12–17].

This naturally raises the question of whether topolog-ical edge states can be protected by localization of thebulk states. Here, we face a new complication not presentin previous examples of localization protected quantumorder: topological bands in fermionic systems cannot becompletely localized by disorder. Rather, the non-trivialband-topology guarantees the existence of an extendedsingle-particle (SP) bulk orbital at some critical SP en-ergy Ec[19] (or more generally a discrete set of such ex-tended orbitals). SP orbitals with energy E near Ec ex-

ξ0

Λ

-Λ

ε

ξ(ε)

ξ=∞

ξ0

ξ(ε)~|ε-εc|-ν

Λ

-Λ

ε

ξ(ε)εcξ0

Λ

-Λ

ε

ξ(ε)

a) b) c)

FIG. 1. Schematic dependence of SP localization length ξon single-particle energy E for a fully-localized system (a),marginally localized system (b), and extended system with amobility edge (c).

hibit diverging SP localization length [18]:

ξ(E) ∼ 1

|E − Ec|ν(1)

where ν is an exponent whose value depends on theparticular system under consideration (see Fig.1). Wedub such systems “marginally localized.” Examples in-clude: disordered integer quantum Hall systems [20 and22] and chiral superconductors, intrinsically topologicalsuperconductors [23], and symmetry protected topolog-ical phases (e.g. topological insulators) with symmetry-preserving disorder [24]. Here, the diverging localizationlength accompanies each change in the topological edge-state structure (e.g. the Chern-number changing quan-tum Hall plateau to plateau transitions). The presence ofat least one extended state is guaranteed by the topologi-cal obstruction to constructing localized Wannier orbitalsin a topological band [20 and 24]. In the vicinity of thecritical energy, the SP localization length diverges as inEq. 1. A diverging SP localization length can occur innon-topological contexts, such as 1D XX spin chains withrandom bond disorder (or equivalently, non-interactingfermion chains with random hopping). Such marginallylocalized systems have properties intermediate betweenlocalized and extended systems, as we shall discuss inSec.II.

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A central issue is whether marginal localization persistsaway from the non-interacting limit. In fully localizedsystems (ξ(E) ≤ ξ0 for all E) there is strong evidencethat localization survives the addition of weak interac-tions even at non-zero energy density [5–7]. In contrast,interactions are expected to delocalize a system whosesingle particle (SP) spectrum contains a mobility edge,as a generic excited state will contain a finite density ofoccupied extended orbitals that thermalize among them-selves and then act as a bath for the localized orbitals.In the presence of a non-zero coupling to a bath, themany body wave functions become thermal, and local-ization is destroyed, even though signatures of proximityto localization may survive in the spectral functions oflocal operators [25 and 26]. Even if extended states onlyarise in a narrow energy range of width W , the resultsof [27] suggest that this narrow band of extended stateswill be sufficient to destroy localization, for any non-zeroW . However, for marginally localized phases, extendedstates arise only at a single energy Ec. In the language of[27], these extended states constitute a bath with vanish-ing bandwidth, and it is not a priori clear whether thiswill be sufficient to destroy localization upon inclusion ofinteractions.

In this paper, we give analytic arguments showing thatlocalization is impossible for arbitrarily weak interactionstrength if νd > 1, where d is the spatial dimension.The breakdown of localization originates from a prolifer-ation of multi-particle resonances between well separatedsingle-particle orbitals. Notably, such breakdown occursin disordered integer quantum Hall systems for whichν ≈ 2.3[21 and 22]. Moreover, in the generic case ofsystems that are governed by the Harris/Chayes bound,νd ≥ 2, and which have a non-zero density of states (e.g.integer quantum Hall systems), our arguments precludethe existence of many body localization at finite energydensity in partially filled bands where the localizationlength diverges at a discrete set of energies.

Our arguments do not exclude many body localizationin systems that evade the Harris bound due to symme-try reasons, and have ν < 1/d. It also does not excludelocalization in systems where the density of states van-ishes sufficiently rapidly at the critical energy. For ex-ample, in a system where the density of states vanishesas |E − Ec|Υ, the breakdown of localization only occursif νd > 1 + Υ. Finally, the Harris/Chayes bound appliesto the critical exponent for the mean localization length,while our argument involves the critical exponent of thetypical localization length, so there remains a possibilityof marginal localization in systems where the mean andtypical localization lengths diverge with very different ex-ponents.

This paper is structured as follows. In Section II wereview the properties of marginally localized phases inthe absence of interactions. In Section III we introducethe model of interest, and the basic approach. In SectionIV we derive a condition for the perturbative instabil-ity of marginal localization. The analysis in Section IV

initially assumes that the wave functions are uniform onlength scales less than a localization length, and decayexponentially on longer length scales. This is an over-simplification, since the wave functions close to criticalityhave multifractal character on length scales short com-pared to the localization length. In Section IV-H we takethis multifractality into account, and demonstrate thatit does not alter our essential results. In Section V wediscuss the consequences of our results, and use the Har-ris/Chayes argument [37 and 38] to rule out stability ofmarginal localization in a wide class of models, includingthe integer quantum Hall effect. We also discuss the pos-sibility of perturbatively stable marginal localization insystems that exploit certain loopholes in our argument,which we discuss.

II. PROPERTIES OF MARGINALLYLOCALIZED PHASES

Before analyzing the effects of interactions, we begin byreviewing the properties of marginally localized phases inthe absence of interactions. In this, and all that follows,we consider the case of only a single critical energy, εcand shift our SP energy scale such that εc = 0. Moreover,we suppose that the SP density of states is regular andnon-zero in the vicinity of εc. We work at partial fillingin the band at a non-zero energy density (i.e. we are notdiscussing ground states or low lying excited states). Inthe quantum Hall context, this amounts to working ata non-zero temperature, but a temperature that is stillmuch less than the cyclotron frequency (allowing us tofocus on a single Landau sub band).

A. Subdiffusive relaxation

Marginally localized states show anomalous subdif-fusive dynamics, as is known from previous studies ofdynamics in disordered quantum Hall systems[39–41].These results are reproduced by the following simple ar-gument. Consider the relaxation of a wave packet Ψ ini-tially localized in the vicinity of R(t = 0) = 0. Thewave packet can be written in the form |Ψ(t = 0)〉 =∑α wα|ψα〉, where |ψα〉 are the single particle states with

energy Eα and amplitude ϕα(r) at a position r. Mean-while, wα are appropriate complex amplitudes that pro-duce a localized wave packet. The wave functions ϕα areall localized, with energy dependent localization lengths,ξ(Eα) as in Eq. 1. This wave packet has the time evolu-tion |Ψ(t)〉 =

∑α wα|ψα〉e−iEαt. We can now straightfor-

wardly evaluate the spatial spreading of the wave packet

3

by examining

R2(t) =

∫ddr

∑α,β

ei(Eα−Eβ)tw∗αwβϕ∗α(r)r2ϕβ(r)

≈∫ddr

∑α

|wα|2ϕ∗α(r)r2ϕβ(r)

≈∑α

|wα|2∑

β:|Eαβ |<1/t

min(Dαt, ξ

2α

)where the second line is valid at long times when thephases between states with |Eαβ | ≡ |Eα − Eβ | � 1/thave effectively randomized, and in going to the thirdline we have assumed that the spreading in each energyinterval is diffusive up to the localization length at thatenergy, and have allowed for the possibility that the dif-fusion “constant” varies with energy. The diffusion con-stant at an energy E may be estimated by setting theThouless energy D(E)/ξ(E)2 equal to the level spacingξ−d(E) at that energy, giving D(E) ∼ ξ2−d. The long-time dynamics will turn out to be controlled by statesnear the critical energy Ec. In dimensions d 6= 2. thesingular behavior of ξ near the critical energy will trans-late into a singular behavior of D(E). However, in twodimensions (the case relevant for quantum Hall), the sin-gularity in ξ will not translate into a singularity in D,and thus we may replace D(E) by a constant value D0,the diffusion constant for the states near the critical en-ergy. Moreover, the detailed initial initial structure of thewave-packet is irrelevant for the long-time asymptotics,which are captured by simply taking |wα| to be roughlyequal amplitude for all energies Eα within a bandwidthΛ of Ec. Under these assumptions, the spreading of thewave-packet follows:

R2(t) =

∫ E(t)

0

dE

ΛD0t+

∫ Λ

E(t)

dE

Λξ2(E) (2)

In the fourth line we have defined the integration limit

E(t) through ξ2[E(t)] ≈ ξ20

(E(t)

Λ

)−2ν

= D0t, i.e. E(t) =

Λ(D0tξ20

)−1/2ν

. The behavior is qualitatively different for

ν < 1/2 and for ν > 1/2. However, provided the Har-ris/Chayes argument[37 and 38] applies (see discussionin Appendix), ν ≥ 1, and only the latter case is relevant.Thus, the dominant contribution comes from low-energystates with diverging ξ(E), i.e. from the 1st term in Eq. 2and from the lower end with E & E(t) in the 2nd termof Eq. 2, and we obtain

R2(t) ∼ tα, α = 1− 1

2ν(3)

Thus the dynamics are characterized by the dynamicalexponent α = 1− 1

2ν intermediate between diffusive (α =1) and localized (α = 0) scaling, and taking the numericalvalue α ≈ 0.78 in the case of integer quantum Hall.

B. Scale dependent conductivity

The sub diffusive spreading Eq. 3 can be interpreted asa length-scale dependent diffusion “constant”: D(L) ∼L−2/(2ν−1). Using the usual Einstein relation betweendiffusion and conductivity, these considerations predictscale dependent DC conductivity

σ(L) ∼ L−2/(2ν−1) (4)

which vanishes in the the thermodynamic limit. Againwe see scaling intermediate between the constant conduc-tivity σex ∼ const of an extended system and the expo-nential scaling σloc(L) ∼ e−L/ξ of a localized system.

C. Fractal Entanglement Scaling

Entanglement scaling provides insight into the char-acter of many-body eigenstates, and provides a concep-tually powerful means to distinguish extended and lo-calized systems. The entanglement entropy of a spatialsub-region of linear size R for an excited energy eigen-state in a localized system comes only from states withina localization length, ξ0, of the boundary and exhibitsboundary-law scaling S(R) ∼ ξ0R

d−1. Here d is thenumber of spatial dimensions. In extended systems, suchboundary law scaling occurs only for the ground-states,whereas excited states typically display volume law scal-ing S(R) ∼ Rd. In this section we analyze the spatialscaling of entanglement entropy in excited states of amarginally localized system, which has not been consid-ered in previous studies.

For a marginally localized system, a fraction f(R) ∼R−1/ν states are extended across the sub-region of sizeR (i.e. have ξ(ε) > R), and make a volume-law typecontribution to entanglement f(R)Rd ∼ Rd−1/ν . Forν > 1, this dominates the area-law contribution fromwell-localized states leading to a scaling:

S(R) ∼ Rd−1/ν (5)

intermediate between volume law and area-law, whichwe dub “fractal” entanglement scaling as it scales likethe boundary of a fractal sub-region of a localized phasewith fractional dimension d− 1/ν.

Marginally localized systems exhibit fractal entangle-ment scaling so long as ν > 1. For ν < 1, the entan-glement entropy becomes effectively area law. When ap-plicable, a Harris-type criterion (see App. A) ν ≥ 2/densures that the fraction of modes which are extendedon the length scale R dominate the entanglement en-tropy in all systems with d < 2. Integer quantum Hallsystems have ν ≈ 2.3, and will exhibit fractal entangle-ment scaling. However Harris/Chayes arguments do notrule out that some marginally localized systems exhibitconventional boundary-law scaling in spatial dimensionsd ≥ 2.

4

Thus, we see that in dynamics, response properties,and the scaling of entanglement entropy, marginally lo-calized non-interacting systems display properties thatare intermediate between traditional localized systemsand delocalized systems. Next, we investigate whetherthis intermediate behavior survives in the presence of in-teractions.

III. MODEL AND APPROACH

The Hamiltonian of interest takes the form H =H0 + V , where H0 is the Hamiltonian of the disorderednon-interacting fermion system (for example Landau lev-els with disorder). This non-interacting Hamiltonian hassingle-particle eigenmodes of energy Eα created by oper-ators ψ†α =

∫ddrϕ∗α(r)c†(r), where c†(r) creates an elec-

tron at position r. Meanwhile, the interaction term takesthe form

V = V∑

α,β,γ,δ

λαβγδψ†αψ†βψγψδ (6)

This represents a short range interaction of strength V ,where λαβγδ =

∫ddrϕ∗α(r)ϕ∗β(r)ϕγ(r)ϕδ(r) are the ma-

trix elements of the interaction in the eigen-basis of H0.We consider the simplest case (illustrated in Fig. 1b)where the SP localization length diverges only at one SPenergy, Ec, and choose our zero of energy to set Ec ≡ 0.The localization lengths for energies close to zero then di-verge as ξ(E) ∼ |E|−ν , such that at any length scale L, anon-zero fraction ∼ L−1/ν of the states appear extended.We take the SP density of states to be a non-zero con-stant in the vicinity of E = 0. For generality we will keepthe spatial dimensionality d and the localization-lengthexponent ν arbitrary, for example d = 2 and ν ≈ 2.3 forthe disordered integer quantum Hall plateau [22].

We now ask whether inclusion of the interaction causesa breakdown of localization. We begin by noting that ifwe attempt to construct the many body eigenstates per-turbatively in weak interactions, then perturbation the-ory for the eigenstates will break down already at O(V )for states involving SP orbitals sufficiently close to themobility edge, irrespective of the value of ν. However,the fact that the eigenstates cannot be perturbativelyconstructed does not imply that localization itself is un-stable - to address the stability of localization we need amore sophisticated diagnostic.

It has been pointed out that fully many-body local-ized systems are locally integrable in the sense that alleigenstates can be completely specified by a set of expo-nentially well-localized conserved, commuting quantitiesor “integrals of motion”[28 and 29]. For non-interactingsystems the integrals of motion are simply the occu-pation numbers of single-particle orbitals. In an MBLphase, these integrals of motion are dressed by inter-actions but (apart from a exponentially small fractionof rare resonances[10]) retain their localized character.In this spirit, one could take the defining property of

marginally localized systems to be the existence of a(nearly) complete set of algebraically well-localized inte-grals of motion. Here, algebraically well-localized meansthat the fraction of integrals of motion with apprecia-ble support on scale larger than R decays as a powerlaw in R, (unlike traditional MBL systems where it de-cays exponentially in R, and also unlike a delocalizedsystem, where the integrals of motion are not localized),and the caveat “nearly” allows for a possible set of non-percolating resonances.

We wish to investigate whether this picture of marginallocalization can be stable in the presence of interactions.To this end, we first introduce some notation. We use|ΦE〉 to denote a (many-body) eigenstate of H0 at an en-ergy E, labelled by a particular set of marginally localizedintegrals of motion, in the sense described above. We use|ΨE〉 to denote a (many-body) eigenstate of H0 + V at

an energy E. We now introduce the T matrix of theproblem by V |ΨE〉 = T |ΦE〉. The matrix elements of Tbetween SP states with energy E and spatial separationR gives the effective matrix element for long range hop-ping. It is well known that the effective hopping matrixelement must fall off with distance faster than 1/Rd, elselocalization is impossible [1 and 31]. Similarly, the ma-

trix elements of T between two particle states with theparticles separated by a distance R gives the effectivematrix element for a long range two-body interaction,the matrix elements between three particle states givethe effective matrix element for a long range three-bodyinteraction etc. Again, there are conditions on how fastthese matrix elements must fall off with R in order for lo-calization to be stable [2, 32, and 33]. Our main strategyin the present work will be to determine how the matrixelements of T fall off with distance R between the parti-cles involved. Hence, we will obtain a condition on ν forbreakdown of localization.

The T matrix is related to V by the Dyson equation[30]

T = V + V1

E − H0 + i0T . (7)

This expression fully specifies the T matrix for statesat a particular energy E. We note that even though Vis short range, T can have long range matrix elementsbecause H0 has eigenstates with arbitrarily long rangematrix elements. Within perturbation theory in smallV , we can approximate the T matrix by the Born series

T = V + V G0(E)V + V G0(E)V G0(E)V + ...(8)

G0(E) =1

E − H0 + i0

This perturbative expression can be represented in stan-dard diagrammatic perturbation theory, where G0 is rep-resented by a (directed) line, and V is represented by avertex with two lines going in and two coming out. Thediagrammatic representation of the T matrix containsonly connected diagrams.

5

IV. DELOCALIZATION IS INEVITABLE IFdν > 1

We use TA,B to denote the matrix element of T be-tween incoming state A and outgoing state B. If TA,Bexceeds the level spacing between states A and B thenthese two states are ‘in resonance,’ and the occupationnumber of state A cannot be an integral of motion. How-ever, any time two states are in resonance, we can definenew states A′ and B′ which are orthogonal linear com-binations of A and B, such that TA′,B′ = 0. The oc-cupation number of A′ and B′ can then be an integralof motion (at least if the states are not resonant withany third state C). However, the new integrals of mo-tion will have support wherever A and B had support.We see therefore that the spatial structure of resonancesis crucial. If resonances only occur between states thathave support at nearby points in real space, then we maybe able to define new integrals of motion which are stilllocalized, if the local resonances do not percolate [31].In contrast, if resonances occur on all length scales, thenthe integrals of motion will not be localized (even alge-braically), and so the system will not be able to supportmarginal MBL.

The question we are interested in asking is thus thefollowing: if A is an eigenstate of H0 with localizationlength ξA, then what is the probability that A is ‘in res-onance’ with another state B a distance R apart, whereR � max(ξA, ξB) (the situation when R < max(ξA, ξB)constitutes a local resonance, which just redefines the lo-calized integrals of motion). If the probability of havinga resonance at a distance greater than R goes to zeroas R → ∞, then we will be able to define algebraicallylocalized integrals of motion with arbitrary precision, in

the sense that all but a fraction ε of the integrals of mo-tion will be localized on a length scale Rc(ε), where Rc(ε)is set by the condition that the probability of having aresonance at a length scale greater than Rc is equal toε. In contrast, if the probability of having a resonanceat a length scale greater than R does not go to zero asR → ∞, then we will not be able to define algebraicallylocalized integrals of motion with arbitrary precision, andthe system cannot support marginal MBL.

We thus want to calculate how the matrix elementsof T scale with distance. We recall that the T matrixis given by the Born series (8). The first order term

V is purely short range. Long range terms first appearat order (V 2). We now examine the how the matrix

elements of T fall off with distance at O(V 2).

A. Hopping resonances

We begin by evaluating the matrix elements of T be-tween well localized SP orbitals ψα and ψη at a separationR � max(ξα, ξη). This can be viewed as the effectivematrix element for hopping at a range R. The medi-ated hopping process at order V 2 is illustrated in Fig. 1.

α β

γ

δ

ηr r'

FIG. 2. Diagrammatic representation of mediated hopping,to lowest order

Thus, we have

Tα,η(R) =∑βγδ

V 2λαβγδλγδβη(Eα + Eβ)− (Eγ + Eδ)

≈ V 2

ξd/2α ξ

d/2η

∑βγδ

exp[−R

(1ξβ

+ 1ξγ

+ 1ξδ

)](ξdβξ

dγξdδ

)(Eα + Eβ − Eγ − Eδ)

exp (−iφβγδ)

= V 2|Eνd/2α Eνd/2η |∑βγδ

Πβγδ(R) (9)

where the sum ranges over intermediate states, (α, β, γ). Since orbitals α, η are well localized we have replaced:λαβγδ ≈ ϕ∗α(0)ϕ∗β(0)ϕγ(0)ϕδ(0), and λαβγδ ≈ ϕ∗λ(R)ϕ∗δ(R)ϕβ(R)ϕη(R). We approximate each of the wave-functions

by the simplistic localized form: |ϕα(r)| ≈ 1

ξd/2α

e−|r−rα| (and similarly for β, λ, . . . ). This assumption ignores the

multi-fractal character of the SP orbitals with ε close to εc, but is expected to reproduce the typical behavior. Themulti-fractal character of the wave functions will be taken into account in Sec. V. For convenience we have defined:

Πβγδ(R) = |EβEγEδ|dνexp [−R (|Eβ |ν + |Eγ |ν + |Eδ|ν)]

(Eα + Eβ − Eγ − Eδ)exp (−iφβγδ) . (10)

Each matrix element contains an overall phase, φβγδ,and the phases are essentially uncorrelated for different

choices of βγδ. Hence the sum over intermediate states

6

acculumates random amplitude and phase characteristicof a random walk in the complex plane of characteristicstep size ≈ |Πβ,γ,δ(R)|. The typical RMS modulus of theeffective hopping is then:

|Tα,η(R)| = V 2|EαEη|νd/2√ ∑βγδβ′γ′δ′

Π∗βγδ(R)Πβ′γ′δ′(R),

= V 2|EαEη|νd/2√∑βγδ

|Πβγδ(R)|2, (11)

where we have assumed that the ‘off diagonal’ terms comewith random phases and cancel out.

We can trade the sum over discrete states for a con-tinuous integral by identifying the appropriate densityof states for intermediate energies Eα,β,δ. Recall thatthe localization length scales as |E|−ν . Thus there are≈ E−dνdE states with appreciable overlap with statesα and η in an energy interval [E,E + dE]. This implies

that we should take∑β →

∫dEβ |E−νdβ |. With the above

rule for translating sums to integrals, we get

〈|Tα,η(R)|〉 = V 2|EαEη|νd/2

√√√√∫dEβdEγdEδ|EβEγEδ|dν

exp[−2R

(Eνβ + Eνγ + Eνδ

)](Eα + Eβ − Eγ − Eδ)2

The exponential scaling of the wave-function amplitudesprovides a cutoff, restricting us to states with |E|β,γ,δ <R−1/ν . Meanwhile, if Eβ,γ,δ are all close to zero (whichmust be the case given the exponential cutoff) then thedenominator is ≈ E2

α, so we obtain

〈|Tα,η(R)|〉V 2

= |Eνd−1α |

√2

∫ R−1/ν

0

dEβdEγdEδ|EβEγEδ|dν

= |Eα|dν−1R−32 (d+1/ν) (12)

This matrix element should be compared to the levelspacing between the states ψα and ψη. If the matrixelement is greater than the level spacing, then the twostates are in resonance i.e. a particle originally in stateψα can readily hop to ψη. If we are to have any hope oflocalization, then the probability of having a resonanceat a length scale R must go to zero as R→∞.

We recall that the effective matrix elements for longrange hopping (at O(V 2)) fall off as R−

32 (d+1/ν). The

probability that we have resonances at a distance R > Rcis then

Pres(R > Rc) =

∫ ∞Rc

Rd−1dR|Tα,η(R)|

δ(13)

where δ ∼ Eνdα is the typical level spacing at energy Eα.Thus, we conclude that the probability of hopping reso-nances at a distance R behaves as

Pres(R > Rc) ∼ V 2|Eα|−1R−(d/2+3/(2ν)c (14)

For any non-zero ε and any non-zero Eα there is anRc such that Pres(R > Rc) < ε. Thus, the effectivelong range hopping falls off sufficiently rapidly that theprobability of having long range hopping resonances van-ishes, and thus the effective long range hopping does notpresent an obstacle to the construction of marginally lo-calized integrals of motion.

B. Flip-flop resonances

Next, we consider the matrix elements of T betweentwo particle states, where the particles in question areseparated by a distance R. Such processes may be repre-sented diagrammatically by processes of the form shownin Fig. 3, and may be interpreted as ‘flip flop’ processeswhich can transport energy over long distances [34]. Theeffective matrix element for this sort of ‘flip-flop’ processscales as

α

β

γ

δ η

μ

r r'

FIG. 3. Diagram illustrating flip-flop process

7

Tαµ,βη(R) =∑γδ

λαβγδλγδµη(Eα + Eβ)− (Eγ + Eδ)

(15)

=V 2

(ξαξβξµξη)d/2

∑γδ

exp[−R

(1ξγ

+ 1ξδ

)](ξdγξ

dδ

)[(Eα − Eβ)− (Eγ − Eδ)]

exp (−iφγδ) (16)

where again φγδ is a random phase. The arguments pro-ceed exactly as for the hopping resonances (except thatthere is one fewer internal variable to sum over), and leadto the result

〈|Tαµ,βη(R)|〉 = V 2 |EαEβEµEη|dν/2

|Eα − Eβ |R−(d+1/ν) (17)

Since the strongest contributions will come from statesα, β, µ, η with similar energies, we take Eγ,δ,β ≈ Eα in thenumerator, and replace the energy denominator, |Eα −Eβ | by minimum level spacing for states of size ξ(Eα) ≈Eνdα , producing:

〈|Tαµ,βη(R)|〉 = V 2Eνdα R−(d+1/ν) (18)

We now ask what is the probability that a particu-lar typical transition has a resonant pair within a dis-tance R. The typical SP level spacing at an energy Eα is∼ ξ−dα ∼ Eνdα . The number of transitions which could bepaired with the transition of interest goes as Rd. Thus,the minimum energy change for flip-flops involving a par-ticular typical transition goes as Eνdα R−d. Meanwhile,the matrix element falls off as V 2Eνdα R−(d+1/ν). Theprobability that a particular transition has a resonantpartner on the length scale R is thus ∼ V 2R−1/ν , whichgoes to zero as R →∞ for any ν. Thus, the probabilityof having a flip-flop resonance will drop below ε on lengthscales R > Rc, where Rc = (V 2/ε)ν . It is instructive tocontrast with the usual MBL case, Fig.1a, where the SPlocalization length is finite at all energies. This situa-tion may be modeled by taking ν → 0, so that all states(except the exact zero energy state) are localized with

localization length of order unity. Taking this limit pre-dicts that for the traditional MBL problem, the lengthscale Rc beyond which the probability of flip-flop reso-nances drops below ε should grow slower than any powerlaw of ε. Meanwhile, the situation with a mobility edgecan be modeled by taking ν → ∞, such that all stateswith E < 1 are delocalized. In this case there will beno finite length scale Rc beyond which the probability offlip flop resonances drops below V 2, indicating instabilityof localization to interactions at finite energy density inpartially filled bands with a SP mobility edge.

C. Flip-flop assisted hopping

Finally, we consider the matrix elements of T betweenthree particle states. This process is represented dia-

α

β γ

δ

ημ

κ

FIG. 4. The flip-flop assisted hopping process

grammatically by Fig.4, and may be interpreted as a‘flip-flop assisted hopping,’ whereby a particle hops fromr1 to r2 by triggering SP transitions at both r1 and r2.We are interested as before in determining how this ma-trix element scales with R = |r1 − r2|. This process hasmatrix element

Tαβµ,γηκ(R) =∑δ

λαβγδλδµηκ(Eα + Eβ)− (Eγ + Eδ)

(19)

=V 2

(ξαξβξγξµξηξκ)d/2

∑δ

exp[− Rξδ

]ξdδ [(Eα − Eβ)− (Eγ − Eδ)]

exp (−iφδ) (20)

where again φδ is a random phase. The arguments pro-ceed exactly as for the flip flop resonances (except thatthere is one fewer internal variable to sum over), and lead

to the result

〈|Tαβµ,γηκ(R)|〉 = V 2 |EαEβEγEµEηEκ|dν/2

|Eα − Eβ |R−

12 (d+1/ν)

(21)

8

a)

...b)

...

r r'

FIG. 5. Some high order diagrams contributing to T

Since the strongest contributions will come from statesα, β, µ, η with similar energies, we take Eγ,δ,β ≈ Eα in thenumerator, and replace the energy denominator, |Eα −Eβ | by minimum level spacing for states of size ξ(Eα) ≈Eνdα , producing:

〈|Tαβµ,γηκ(R)|〉 = V 2E2νdα R−

12 (d+1/ν) (22)

There are∼ Rd sites within a distanceR of ψα to whicha particle could hop via this ‘flip-flop assisted’ hoppingprocess. Thus, the minimum level spacing for such a pro-cess falls off as R−d. Meanwhile, the matrix element forthis process falls off only as R−

12 (d+1/ν). Thus, the prob-

ability that there is no ‘assisted hopping’ resonance ata length scale R vanishes at long distances as a powerlaw of R for νd > 1 (for νd = 1 it vanishes logarithmi-cally with R). Thus, for νd ≥ 1, a particle can alwaysfind a (distant) resonant site to hop to, with some assis-tance from flip flops. This establishes that localization isimpossible in the presence of interactions if

dν ≥ 1 (23)

This is the main result of this section, and concludesour discussion of the matrix elements of T at O(V 2) (we

note that T has no matrix elements between four particlestates at O(V 2) because T must be represented by a con-nected diagram). Next, we discuss whether we can geta more stringent condition for breakdown of localizationat higher orders in V .

D. Higher orders in V

We now discuss whether a more stringent condition onbreakdown of localization could be obtained by going tohigher order in V and considering diagrams, e.g. of thesort illustrated in Fig.5. We will conclude that the an-swer is no, and that (23) is the criterion for perturbativebreakdown of marginal localization. We begin by notingthat the diagrammatic representation of the Born series(8) contains vertices, internal lines (which begin and endon a vertex), and external lines (which are connectedto vertices only at one end). At O(V n) in perturbationtheory, we will have n vertices, m internal lines and pexternal lines, where 2m+ p = 4n. Since only connecteddiagrams contribute to the T matrix, every vertex must

be connected to at least one internal line, and all but twoof the vertices must be connected to at least two internallines. This gives the constraint m ≥ (n− 1).

Now, the R scaling depends only on the number ofinternal lines m. After summing over internal variablesin a rms sense, we find that the R scaling has the form

T (R) ∼

√√√√[∫ R−1/ν

dEE−νd(Eνd)2

]m(24)

where the first factor E−νd comes about when we changethe sum over states to an integral, and the second factor(Eνd)2 comes from the normalization factors associatedwith the internal lines. This gives

T (R) ∼ R−m2 (d+1/ν) (25)

Now, if we evaluate the matrix element of T that con-nects a particle at a given position to particles at n otherpositions within a ball of radius R (i.e. if we look at di-agrams containing n ≥ n+ 1 vertices that are connectedto at least one external line), then the minimum levelspacing for such processes will fall off as ∼ R−nd. If thislevel spacing falls off faster than the corresponding ma-trix element, then the probability of having resonanceswill approach one at large R, so that localization will beunstable. The condition for delocalization thus takes theform

nd >m

2

(d+

1

ν

)(26)

Thus, to maximize our chances of delocalization, weshould maximize n and should minimize m, subject tothe contraints we have already identified n + 1 ≤ n andm ≥ n − 1. Substituting n = n − 1 and m = n − 1into the above formula, we find that the condition fordelocalization becomes

(n− 1)d >n− 1

2

(d+

1

ν

)⇒ dν > 1 (27)

Thus, the tightest constraint that we could possibly getwithin perturbation theory is the constraint (23) com-ing from the flip-flop mediated hopping process i.e. de-localization occurs if dν > 1. Meanwhile, for dν < 1,the probability of resonances vanishes at large R at allorders in perturbation theory, so that marginal localiza-tion appears to be perturbatively stable. For dν = 1the probability of having a resonance, calculated in themanner of (13), grows logarithmically with R, suggest-ing that the system should be delocalized (this is alsoconsistent with results on non-interacting models withcritically long range hopping [31]).

E. Systems with a vanishing critical density ofstates

The result (23) is slightly modified if the density ofstates vanishes as EΥ in the vicinity of the critical energy.

9

In this case, the integrals over energy come with a factorof EΥ, and the matrix elements with m internal lines areof order

T (R) ∼ R−m2 (d+(1+Υ)/ν) (28)

In this case similar arguments establish that delocaliza-tion occurs if

νd ≥ 1 + Υ (29)

The situation where the density of states diverges at thecritical energy can also be treated within the above for-malism, by taking Υ < 0.

F. An intuitive explanation for these results

We now draw attention to a very simple and intuitiveexplanation for the delocalization criterion (23). On anylength scale R, there are states with energies |E| < R−1/ν

that look effectively extended. Meanwhile, there areξd(E) states at an energy E that have support at a par-ticular position r. Thus, the number of states to whichwe can couple with a local interaction which are extendedon the scale R is

N(R) ∼∫ R−1/ν

0

dEEΥξd(E) =

∫ R−1/ν

0

dEEΥ−νd (30)

If νd < 1 + Υ then the ‘bath’ of extended states to whichwe can locally couple contains a finite number of states(and the number of states in the bath goes at zero as thelength scale R → ∞). In this case, localization can beperturbatively stable. In contrast, if νd > 1 + Υ, thenthe above integral diverges near E = 0, such that the‘bath’ to which we can locally couple contains an infinitenumber of extended states at any length scale R. In thiscase, the algebraically delocalized states near the criticalenergy form a good bath, and are able to thermalize thesystem by mediating long range interactions that fall offsufficiently slowly with R, as we have discussed.

G. Characteristic length- and time- scales fordelocalization

We now estimate the length scale and timescale forthe resonances that give rise to a breakdown of local-ization. Recall that the T matrix elements betweenstates separated by a distance R at order n + 1 in per-turbation theory scale as V n+1R−n(d+1/ν)/2, while thelevel spacing scales as R−nd. The length scale Rc(n)on which resonances appear at order n + 1 in pertur-bation theory is the length scale at which the matrixelements and the level spacings become comparable i.e.

V n+1R−n(d+1/ν)/2c = R−ndc . This happens at a critical

length scale Rc, where

Rc(n) ∼ V −2νdν−1

n+1n (31)

By inspection, the smallest Rc comes from n → ∞ forwhich

Rc ∼ V −2ν/(dν−1). (32)

The scaling of Rc in Eq. 32 controls, for example, thelength scale at which the longitudinal conductivity ceasesto follow (4) and saturates instead to a non-zero constant.Remarkably, even though breakdown of perturbation the-ory occurs already at second order in perturbation theory,the shortest length scale on which resonances appear iscontrolled by high orders of perturbation theory.

We can also estimate the timescale on which break-down of localization occurs. Processes at order n + 1cause a breakdown on length scales Rc(n), given by(31). The matrix element for these processes is of or-der V n+1Rc(n)−n(d+1/ν)/2 = Rc(n)−nd, and the levelspacing is Rc(n)−nd, so that the density of final statesis Rc(n)d. A straightforward application of the GoldenRule implies that the timescale on which breakdown oflocalization happens due to processes at n + 1 order inperturbation theory is

tc(n) ∼ Rc(n)nd = V −2νd(n+1)νd−1 (33)

Thus we see that the shortest timescale on which break-down occurs is controlled by n = 1 (i.e. by processes atsecond order in perturbation theory), and is of order

tc ∼ V −4νdνd−1 (34)

For the case of integer quantum Hall, with ν = 2.3 andd = 2, Rc ∼ V −1.3 and tc ∼ V −2.5. We note that thecrossover timescales Rc and tc both diverge as V → 0,recovering marginal localization in the non-interactinglimit (V = 0).

So far we have suppressed temperature-dependent oc-cupation factors in estimating the T-matrix amplitudes.The breakdown of marginal localization happens (for aclosed quantum system) because the ‘internal heat bath’presented by the states with diverging localization lengthallows the system to thermalize. At low temperature, thecoupling to this internal bath is suppressed by a factorof exp(−∆/T ), where ∆ is the detuning of the chemi-cal potential from the critical energy - i.e. the crossoverscales Rc and tc will be enhanced by an ‘Arrhenius’ factor∼ exp(c∆/T ), where c is a positive numerical pre factorthat depends on ν and d. Thus, the crossover scales willalso diverge in the zero temperature limit, so that thezero temperature system will indeed be localized, as longas the Fermi level is not at the critical energy.

The emergence of a new length scale and timescale inthe interacting problem suggests that the scaling behav-ior near the zero temperature quantum critical point [22]may be richer for the interacting system than for the non-interacting system. In particular, the presence of an ad-ditional interaction related length scale Rc may help ex-plain why the experimentally observed longitudinal con-ductance is not temperature independent at the critical

10

point [22] - the longitudinal conductance could dependon the ratio of Rc to the thermal length ~/kT . Addition-ally, if one examines the scaling of e.g. the longitudinalconductance with temperature at a non-zero detuning ∆from the critical point, then the coupling to the ‘internal’heat bath will provide a mechanism for a thermally acti-vated scaling of the conductance, with an activation gapproportional to the detuning from the critical point. Ofcourse, any real experimental system will also be coupledto an ‘external’ heat bath (e.g phonons), which will allowfor variable range hopping at any non-zero temperature,and this variable range hopping contribution to the con-ductance will dominate over the ‘activated’ contributionat the lowest temperatures. However, if one observed aregime where the low temperature longitudinal conduc-tance was activated, with an activation gap proportionalto the detuning from criticality, then this may be tenta-tively identified as evidence for the ‘delocalization due tothe internal bath’ discussed in this paper. An additionalcomplication is that we have considered short range in-teractions, while in real quantum Hall systems there arealso Coulomb interactions (unless screened by a nearbymetallic gate). The inclusion of long-range Coulomb in-teractions may introduce additional corrections to scalingthat are not captured by the present theory.

H. Multifractality

Thus far we have assumed that the wave functionswere uniform on length scales short compared to ξ. Infact, this is not the case, because close to criticality thewave functions exhibit multi fractal behavior on lengthscales less than ξ. It is known [35 and 36] that for wavefunctions close to criticality, the disorder averaged twopoint correlations fall off as 〈ψ∗α(0)ψβ(0)ψ∗β(r)ψα(r)〉 ∼

1ξ2d (ξ/r)η for r < ξ, where η > 0 is a multi fractal ex-

ponent. This suggests that the wave functions actuallyhave higher amplitudes on scales r � ξ than would beexpected based on the ‘plain vanilla’ model we were work-ing with thus far. This will tend to enhance the matrixelements of the T matrix over and above the estimatesabove. Thus, the condition νd > 1+Υ may be viewed asa lower bound condition for delocalization. If we take intoaccount the effects of multifractality in a simple minded

way by attaching a factor of (ξint(E)/R)η/2

to each in-ternal line, then we will conclude that the contributionto the T matrix from a diagram with m internal lines(after rms summation over internal variables) will be

T (R) ∼

√√√√[∫ R−1/ν

dEEΥ−νd(Eνd)2

(E−ν

R

)η]m∼ R−

m2 (d+ 1+Υ

ν ) (35)

which is independent of η - i.e. if we take multifractalityinto account in this simple minded way then the thresh-old for delocalization is unchanged, and is νd ≥ 1 + Υ.

A more sophisticated treatment of multi-fractality mayreduce the critical value of ν somewhat, but (as we havediscussed), multifractality can only make it easier to getdelocalization.

V. DISCUSSION AND CONCLUSION

We have discussed marginally localized systems, whichfor partially filled bands at non-zero energy density arecharacterized in the non-interacting limit by sub diffu-sive relaxation (3), a scale dependent DC conductivitythat vanishes in the thermodynamic limit (4) and an en-tanglement entropy of the excited eigenstates that is in-termediate between area law and volume law (5). Wehave shown that such systems are generically unstable tointeractions, since interactions trigger a breakdown of lo-calization for any ν > 1/d. We further note that, for mostsystems, general arguments ([37 and 38] and Appendix)place a lower bound, ν ≥ 2/d, on the localization lengthexponent. Thus, we have shown that marginally localiza-tion is generically unstable to interactions, for partiallyfilled bands at non-zero energy density.

We therefore predict that in a marginally localized sys-tem such as integer quantum Hall, the inclusion of arbi-trarily weak short range interactions will cause the entan-glement entropy in the excited many body eigenstates tobecome volume law, instead of the fractal scaling shownin (5). We also predict that in two dimensional weaklyinteracting marginally localized systems, the scale depen-dent conductivity at nonzero energy density will follow(4) on small length scales, but will saturate to a non-zeroconstant on large length scales. Similarly, relaxation insuch marginally localized systems should be sub-diffusiveaccording to (3) for short times, but should cross overto diffusive relaxation on long timescales. We have es-timated the crossover length scales and timescales, andhave also discussed in Sec.IV-G the interaction inducedcorrections to scaling in the vicinity of the zero tempera-ture quantum critical point. These ideas may be directlytested in experiments in cold-atomic systems and possi-bly also in solid-state materials for which the electron-phonon interactions are negligible, so that the electronicsystem well approximates an isolated quantum system.We speculate in particular that disordered graphene inthe quantum Hall regime may provide a playground totest these ideas, due to its weak electron-phonon couplingand isolation from any substrate in suspended devices.

While our arguments were for short range interac-tions, the question of whether MBL can survive with longrange interactions has also been recently studied [33]. Itwas claimed there that full MBL is perturbatively sta-ble against weak interactions as long as the interactionsfall off with distance faster than 1/rp, where p is a crit-ical power that depends on the model in question. Forinteractions that fall off faster than 1/rp, our argumentscan be straightforwardly generalized to demonstrate thatmarginal localization is perturbatively unstable. The sit-

11

uation with truly long range interactions (which fall offslower than 1/rp), is beyond the scope of the presentwork.

We note that there are three classes of short range in-teracting system that may evade our arguments. If thedivergence in the localization length is accompanied by adensity of states (DOS) which vanishes as EΥ, our argu-ment only predicts delocalization for dν > 1 + Υ. Givena sufficiently rapid vanishing of the DOS at the criticalenergy, marginal localization could be perturbatively sta-ble. Additionally, in systems where the critical energy ispinned (e.g. by symmetry) to a particular value and is lo-cally insensitive to disorder, the Harris criterion does notapply and we can have ν < 2/d (see [42] and Appendix).A particular example of this is models of particle-holesymmetric localization in one dimension [23], where thedivergence in the localization length is only a logarithmicfunction of energy. In this case our analysis would suggestthat marginal localization could be perturbatively stable.Finally, our arguments establish that marginal localiza-tion is unstable for ν > 1/d, where ν is the critical expo-nent for the localization length of typical states. However,

the Harris/Chayes criteria only constrain the critical ex-ponent for the mean localization length νmean ≥ 2/d. Forinfinite randomness critical points, νtypical and νmean canbe markedly different[43]. Thus, it is conceivable thatthere may be systems with infinite randomness criticalpoints where νmean ≥ 2/d (satisfying Harris/Chayes),but νtypical < 1/d, so that marginal localization couldbe perturbatively stable. Investigations of whether suchmarginally localized systems can be stable to interactionswould be a fruitful topic for future work.

VI. ACKNOWLEDGEMENTS

We thank David A. Huse, Siddharth Parameswaran,S.L. Sondhi, Romain Vasseur, Ashvin Vishwanath, RavinBhatt, and Norman Y. Yao for helpful conversations. Wealso thank David A. Huse and Sarang Gopalakrishnan fora critical reading of the manuscript. RN is supported bya PCTS fellowship.

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Oganesyan14 R. Vosk and E. Altman, arXiv: 1307.325615 B. Bauer and C. Nayak, J. Stat. Mech. (2013) P0900516 Y. Bahri, R. Vosk, E. Altman and A. Vishwanath

arXiv:1307.4192 (2013).17 A. Chandran, V. Khemani, C.R. Laumann and S.L.

Sondhi, arxiv: 1310.294618 We note that we are interested in how the localization

length of a typical state diverges with energy. For infiniterandomness critical points, the typical and mean localiza-tion lengths may well scale differently. In such a cast, ν inour arguments should be understood as νtypical.

19 B.I. Halperin Phys. Rev. B 25, 2185 (1982)

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25 R. Nandkishore, S. Gopalakrishnan and D. A. Huse, arXiv:1402.0686

26 S. Johri, R. Nandkishore and R.N. Bhatt, arXiv: 1405.551527 S. Gopalakrishnan and R. Nandkishore, arXiv: 1405.103628 M. Serbyn, Z. Papic, and D. A. Abanin, Phys. Rev. Lett.

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M. Muller, E.A. Demler and M. D. Lukin, arxiv: 1311.715134 Permuting the arrows in the diagram also gives a ‘long

range pair hopping’ process whereby two particles at ener-gies ±E and position r1 hop to two states at energy ≈ ±Eand position r2 with |r2 − r1| = R. However, this ma-trix element scales with R in the same way as the flip flopprocess, and so we do not discuss it separately.

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Appendix A: Harris/Chayes bounds for LocalizationLength Exponent

The generating function for two-body Green’s func-tions with fixed energy ε of a non-interacting quantumsystem in d spatial dimensions can be written in termsof a partition function with a d-dimensional action:

Z(ε) =

∫D[ψ, ψ]eiψ(ε+i0+−H)ψ (A1)

where H is the single-particle Hamiltonian including ran-dom disorder. A diverging SP localization length corre-sponds to a critical value of εc for which Z(ε) exhibitsa phase transition. Harris’s argument, which we brieflyreview below, can then be applied to this d-dimensionaltheory to constrain the correlation length exponent ν.

Within a region of size L, the disorder potential givesa random shift to the average value of ε within the region

of order δ(L) ∼ L−d/2. Then, in order to determine theaverage value of ε − εc to accuracy δ, one needs to lookat lengthscale L ∼ δ−2/d. On the other hand, if the tran-sition occurs inside the single particle band, one shouldbe able to determine on which side of the transition thesystem lies by examining a sub-system with size L muchbigger than the correlation length ξ(ε) ∼ |ε− εc|−ν . To-gether these considerations require that the random shiftin average ε in a subsystem of size ξ(ε) goes to zerono slower than than the distance to the critical pointlimε→εc

δ (ξ(ε)) /|ε− εc| = 0, or:

dν ≥ 2, (A2)

the celebrated Harris criterion.

This argument applies only if the critical energy canlocally be shifted by disorder [42]. In situations wherethe extended states are pinned at a particular energyindependent of disorder, perhaps by a symmetry, thenthe Harris bound does not apply, and the system canhave νd < 2. An example of this situation is the case ofparticle-hole symmetric localization [23], where the de-localized states sit exactly at E = 0 because of particlehole symmetry independent of the strength of the (ran-dom hopping) disorder. In this case, the divergence ofthe localization length is only a logarithmic function ofthe energy (i.e. slower than any power law), and our ar-gument suggests that marginal localization could be atleast perturbatively stable.