Many-body localization in Fock-space: a local perspective
MMany-body localization in Fock-space: a local perspective
David E. Logan and Staszek Welsh
Department of Chemistry, Physical and Theoretical Chemistry,Oxford University, South Parks Road, Oxford, OX1 3QZ, United Kingdom (Dated: January 9, 2019)A canonical model for many-body localization (MBL) is studied, of interacting spinless fermionson a lattice with uncorrelated quenched site-disorder. The model maps onto a tight-binding modelon a ‘Fock-space (FS) lattice’ of many-body states, with an extensive local connectivity. We seek tounderstand some aspects of MBL from this perspective, via local propagators for the FS lattice andtheir self-energies (SE’s); focusing on the SE probability distributions, over disorder and FS sites. Aprobabilistic mean-field theory (MFT) is first developed, centered on self-consistent determinationof the geometric mean of the distribution. Despite its simplicity this captures some key featuresof the problem, including recovery of an MBL transition, and predictions for the forms of the SEdistributions. The problem is then studied numerically in 1 d by exact diagonalization, free fromMFT assumptions. The geometric mean indeed appears to act as a suitable order parameter forthe transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have auniversal form, with long-tailed L´evy behavior as predicted by MFT. In the delocalized phase forweak disorder, SE distributions are clearly log-normal; while on approaching the transition theyacquire an intermediate L´evy-tail regime, indicative of the incipient MBL phase. PACS numbers: 71.23.-k, 71.10.-w, 05.30.-d
I. INTRODUCTION
Sixty years ago Anderson famously discovered thephenomenon of localization, specifically in single-particlesystems. While the importance of disorder in interactingsystems has long been appreciated, traditional studyof it has focused on ground state quantum phases andtheir transitions. More recently, stimulated in part bybasic issues relating to thermalization (or its absence) inisolated quantum systems, attention has turned to studyof highly excited quantum states, and the phenomenonof many-body localization (MBL) occurring at finiteenergy density above the ground state.Over the last decade or so the MBL problem has at-tracted great interest, and a rich understanding of ithas begun to emerge (for a review see e.g. [22]). Power-ful diagnostics have been deployed to understand prop-erties of both the ergodic and MBL phases, includinglevel statistics and related measures, and en-tanglement entropies and spectra. TheMBL phase itself has naturally attracted particular at-tention, highlights including an existence proof for thephase, the description of it in terms of local integrals ofmotion, and an emerging understand-ing of how MBL states exhibit broken symmetries andtopological order. Numerical methods have of courseplayed a key role, from more or less ubiquitous exactdiagonalization studies, to a variety of RG-based meth-ods.
A range of relevant models have like-wise been studied, notably the ‘standard’ spinless fermionor random XXZ models, but including also quasiperiodicmodels, and MBL systems subject to periodicdriving.
An understanding of MBL is never-theless still in relative infancy, with abundant potentialfor new insights to emerge. Here we take a rather different approach to the prob-lem; focusing for specificity on the standard model of in-teracting spinless fermions with quenched site-disorder,for a lattice of N ≡ L d sites occupied by N e fermionsat non-zero filling ν = N e /N . As reprised in sec. II,the model can be mapped exactly onto a tight-bindingmodel (TBM) on a ‘Fock-space (FS) lattice’ of dimen-sion N H = N C N e ( ≡ (cid:0) NN e (cid:1) ), each site I of which corre-sponds to a disordered, interacting many-body state withspecified fermion occupancy of all real-space sites, and isthus associated with a FS ‘site-energy’; while FS sites areconnected by the one-electron hopping matrix element ofthe original Hamiltonian H . The FS lattice is of course acomplex network, with the associated TBM verydifferent from that typically encountered in one-body lo-calization. The local connectivity of a FS lattice site forexample – the number of FS sites to which it is connectedunder hopping – is typically extensive ( ∝ N ); and whilethe N site-energies of the underlying real-space latticesites are independent random variables, the FS site en-ergies are certainly not (there being exponentially manyof them, N H = N C N e ).Our aim here is to exploit the above mapping, seekingto obtain some insight into aspects of MBL. To that endwe consider the local (i.e. site-diagonal) propagators forthe FS lattice, appropriately rescaled in terms of the stan-dard deviation of the eigenvalue spectrum (secs. II B,III);focusing specifically on their associated local (or Feen-berg) self-energies. An approach of this general ilkhas a long history in studies of one-body localization, and in an early approach to MBL in the context of vi-brational energy flow, but has not to our knowledgebeen considered in the recent era of MBL. The imaginarypart ˜∆ I of the local self-energy is of central importance,as its behavior discriminates cleanly between delocalized a r X i v : . [ c ond - m a t . s t r- e l ] J a n and MBL phases, being finite with probability unity forthe former and vanishingly small for an MBL phase. Wethus study its probability distribution, over disorder real-izations and FS sites; noting at the outset that its meanvalue (cid:104) ˜∆ I (cid:105) – which amounts to Fermi’s golden rule – isnon-zero throughout both phases (sec. III), and as suchis irrelevant as a diagnostic for the MBL transition.A probabilistic mean-field theory is first developed(secs. IV,V), treated at the level of the second-orderrenormalized perturbation series for the self-energy;and centering on a self-consistent determination of thegeometric mean of the distribution, which acts as an ef-fective order parameter for the transition. Despite itssimplicity this appears to capture some key features ofthe problem, including recovery of an MBL transition inthe thermodynamic limit – with some physical under-standing of how and why this arises, given the extensivelocal connectivity of the FS lattice – and predictions forthe self-energy distributions in each phase, and their evo-lution with disorder, interaction and filling fraction.Free from the assumptions entering the mean-field the-ory, in sec. VI the problem is then studied numerically in1 d via exact diagonalization. Results arising broadly con-cur well with those from the mean-field approach, par-ticularly in the MBL phase. The geometric mean in-deed appears to act as a suitable order parameter for thetransition. Throughout the MBL phase the appropri-ate self-energy distribution is likewise confirmed to havea universal form, with the long-tailed L´evy behavior aspredicted by the mean-field theory; and with further un-derstanding provided by considering the single-particlelimit, where much larger system sizes can be studied, andfor which exact results can be obtained. In the delocal-ized phase for weak disorder, self-energy distributions arefound to be log-normal; and while remaining unimodalwith increasing disorder, on approaching the transitionthey appear to acquire an intermediate L´evy tail regime,indicative of the incipient MBL phase, before crossingover to an exponentially damped inverse Gaussian form.
II. MODEL AND BACKGROUND
The spinless fermion model considered is the standardone, H = H W + H t + H V (1a)= (cid:88) i (cid:15) i ˆ n i + (cid:88) (cid:104) ij (cid:105) t ( c † i c j + c † j c i ) + (cid:88) (cid:104) ij (cid:105) V ˆ n i ˆ n j (1b)(where ˆ n i = c † i c i ); here considered on a d -dimensionalhypercubic lattice of coordination number Z d = 2 d . Thehoppings ( t ) and interactions ( V ) are nearest neighbor(NN), and (cid:104) ij (cid:105) denotes distinct NN pairs. In the disor-dered H W , the site energies { (cid:15) i } are independent randomvariables, with a common distribution P ( (cid:15) i ) (chosen tohave zero mean). The lattice has N sites and contains N e fermions; and we are interested in the thermodynamic limit where both N ≡ L d and N e → ∞ , holding the filling ν = N e /N fixed and non-vanishing. For the particularcase of d = 1 the model is equivalent under a Jordan-Wigner transformation to the random XXZ model (withtotal spin S Z tot ≡ ( ν − ) N ).The dimension of the associated Fock-space (FS) is N H = N C N e , and is exponentially large in the numberof sites ( N H ∝ exp[ s ∞ N ] with s ∞ = − [ ν ln ν + (1 − ν ) ln(1 − ν )] the configurational entropy per site). TheHamiltonian may be recast equivalently as an effectivetight-binding model on a ‘FS-lattice’ of N H sites { I } , H = (cid:88) I E I | I (cid:105)(cid:104) I | + (cid:88) I,J ( J (cid:54) = I ) T IJ | I (cid:105)(cid:104) J | . (2)The N H basis states {| I (cid:105)} are the eigenstates of H = H W + H V , viz. | I (cid:105) = with occupation number n ( I ) i = 0 or1 only for each real-space site i (such that (cid:80) i n ( I ) i = N e for any | I (cid:105) ). Corresponding FS site-energies follow from H | I (cid:105) = E I | I (cid:105) as E I = (cid:88) i (cid:15) i n ( I ) i + V r I : r I = (cid:88) (cid:104) ij (cid:105) n ( I ) i n ( I ) j (3)where r I is thus defined (and simply counts the totalnumber of occupied NN real-space pairs in | I (cid:105) ). FS sitesare connected under the hopping, T IJ = (cid:104) I | H t | J (cid:105) , with | T IJ | = t when non-zero ( T IJ is either + t or − t for gen-eral d ≥
2, its sign depending on the configuration offermions in | I (cid:105) and | J (cid:105) ; while T IJ = t for all connectedFS sites in the d = 1 open chain). The number of FSsites to which any given I is connected defines the localcoordination number, Z I . This is readily shown to begiven by Z I = 2( dN e − r I ), so follows directly from r I .Eigenvalues and eigenfunctions of H are denotedby E n and | Ψ n (cid:105) . The eigenvalue spectrum D ( ω ) = N − H (cid:80) n δ ( ω − E n ) is normally distributed, with an ex-tensive mean E ∝ N and a standard deviation µ E ∝ √ N ;and it is states prescribed by this energy regime on whichwe focus throughout. Such states correspond in the com-monly used sense to infinite-temperature states (and in-clude all but an exponentially small fraction of eigen-states).Given the mapping to an effective TBM on a FS lattice,the same questions can be asked as arise for a single-particle disordered TBM, including the central one ofwhether eigenstates of some given energy are localizedor extended. As for one-body localization (1BL), theanswer to this question resides in principle in the dis-tribution, over FS sites and disorder realizations, of thesquared amplitudes | A nI | = |(cid:104) I | Ψ n (cid:105)| in the eigenfunc-tion expansion | Ψ n (cid:105) = (cid:88) I A nI | I (cid:105) (4)(with (cid:80) I | A nI | = 1 = (cid:80) n | A nI | , normalization). Thismerits elaboration, as there are both differences and sim-ilarities between the MBL and 1BL cases.In a one-body problem the site label I refers of courseto a single real-space site ( | I (cid:105) ≡ | i (cid:105) = c † i | vac (cid:105) , with theFock-space dimension simply N H ≡ N C = N ). Here, anextended state has support on O ( N ) sites, with a typi-cal | A nI | on the order of ∼ /N (recall (cid:80) I | A nI | = 1);while a localized state has support on a finite number n s of sites, with a typical | A nI | thus O (1 /n s ), likewisefinite. The fraction of sites on which a extended statehas support is thus O (1) in the thermodynamic limit N → ∞ , while the fraction on which a localized statehas support vanishes. In the many-body case, anextended state analogously has support on O ( N H ) sites,with a typical | A nI | ∼ /N H . A many-body localizedstate has however support on O ( N α H ) sites with α < | A nI | ∼ N − α H ; in otherwords it has support on a number of FS sites of order N α H ∝ e αs ∞ N which is exponentially large in N , in con-trast to a finite number in the corresponding 1BL case.Nevertheless, the fraction ∼ N α − H of FS sites on whichan MBL state has support again vanishes in the thermo-dynamic limit, while the corresponding fraction for anextended many-body state remains O (1); which is in ev-ident parallel to the 1BL case. Bounded as they are inthe thermodynamic limit, whether these fractions are fi-nite or vanish is one reflection of the distinction betweenextended and localized states, for both MBL and 1BLproblems.These simple qualitative considerations naturally haveimplications for e.g. the first participation entropy, onediagnostic of the amplitude distributions, defined fora state | Ψ n (cid:105) by S pe1 = − (cid:80) I | A nI | ln | A nI | . For an ex-tended many-body state, taking | A nI | ∼ /N H on all N H sites gives S pe1 = ln N H ; while for an MBL state,taking | A nI | ∼ N − α H on N α H sites, gives S pe1 = α ln N H ( α < Note furtherthat, since S pe1 ∝ ln N H grows with system size, it is S pe1 / ln N H ∝ S pe1 /N which remains bounded in the ther-modynamic limit. This illustrates that relevant physicalquantities are liable to require suitable rescaling with sys-tem size to obtain a well defined thermodynamic limit(further examples of which will arise below). A. Basic distributions
It proves helpful in the following to have some under-standing of the FS lattice in a statistical sense; specifi-cally the distributions, over both FS sites and (where ap-propriate) disorder realizations, of: r I (eq. 3) and hencethe FS coordination number Z I = 2( dN e − r I ) (whichare disorder-independent), as well as the distributions ofFS site energies {E I } , and of the eigenvalues { E n } of H .These are considered in [59], from which we now recaprequired results.In the thermodynamic limit of interest all such dis-tributions are normal, with extensive means O ∝ N and standard deviations µ O ∝ √ N . The mean of r I is r = νdN e = ν dN , so the average coordination number Z = Z d (1 − ν ) N e = 2 ν (1 − ν ) dN. (5)That Z ∝ N is physically obvious, but begs the questionof how Fock-space localization survives the thermody-namic limit, itself considered in secs. IV ff. As for themean FS site energy ( E ) and eigenvalue ( E ), they coin-cide and are given by E = E = V r = V ν dN. (6)The variance of r I is µ r = [ ν (1 − ν )] dN (whence µ Z =4 µ r for the coordination number). That for the FS siteenergies (eq. 3) is µ E = µ W + µ r , with µ W the standarddeviation of the disorder term in eq. 3 ( ≡ H W ), given by µ W = √ N (cid:2) ν (1 − ν ) (cid:104) (cid:15) (cid:105) (cid:3) (7)( (cid:104) (cid:15) (cid:105) = (cid:82) d(cid:15)P ( (cid:15) ) (cid:15) ); such that µ E = √ N (cid:2) ν (1 − ν ) (cid:104) (cid:15) (cid:105) + V [ ν (1 − ν )] d (cid:3) . (8)Finally, the eigenvalue variance is given simply by µ E = µ E + t Z. (9) B. Energy rescaling
The normalized eigenvalue distribution is just the den-sity of states (DoS), D ( ω ) = N − H (cid:80) n δ ( ω − E n ) ( ≡ N − H (cid:104) (cid:80) n δ ( ω − E n ) (cid:105) (cid:15) by self-averaging). As above, itis the Gaussian D ( ω ) = 1 √ πµ E exp (cid:16) − [ ω − E ] µ E (cid:17) , (10)which we add is very well captured by exact diagonal-ization on the modest system sizes amenable to numerics. D ( ω ) is obviously N -dependent, from both E ∝ N andbecause its standard deviation µ E ∝ √ N . As such itis natural to refer energies relative to the mean, and torescale them in terms of the spectral width µ E via˜ ω = ( ω − E ) /µ E ≡ ( ω − E ) /µ E . (11)With this the DoS ˜ D (˜ ω ) is a standard normal distribution(zero mean and unit variance),˜ D (˜ ω ) = 1 √ π exp (cid:0) − ˜ ω (cid:1) (12)with no explicit N -dependence. This rescaling provescentral to our perspective on MBL, as will be seen in thefollowing; and while so far motivated physically is in factrequired on general grounds. III. LOCAL PROPAGATORS ANDSELF-ENERGIES
We turn now the FS site propagators, and in particularthe local (site-diagonal) propagator, the associated self-energy of which is our primary focus.The site-dependent propagators for the FS lattice, G JI ( ω ) ( ↔ G JI ( t ) = − iθ ( t ) (cid:104) J | e − iHt | I (cid:105) ), are given by G JI ( ω ) = (cid:104) J | ( ω + − H ) − | I (cid:105) with ω + = ω + iη and η ≡ H has the TBM form eq. 2, it followsdirectly that( ω + − E I ) G IJ ( ω ) = δ IJ + (cid:88) K T IK G KJ ( ω ) , (13)which generates the familiar locator-series expansion forthe propagators. In particular, the local propagator G I ( ω ) ≡ G II ( ω ) is G I ( ω ) = (cid:80) n | A nI | / ( ω + − E n ); fromwhich follows the local DoS, D I ( ω ) = − π Im G I ( ω ) = (cid:80) n | A nI | δ ( ω − E n ), i.e. the local density of eigenvalueswhich overlap site I . These in turn are related to the total DoS – the Gaussian eq. 10 – by D ( ω ) = N H (cid:80) I D I ( ω ).The local propagator has been known since Ander-son’s original work to be of particular importance inone-body localization; and that is also true in the MBLcontext, given the mapping to the effective TBM eq. 2.It is most effectively analyzed in terms of the Feenbergself-energy S I ( ω ) = X I ( ω ) − i ∆ I ( ω ), and in partic-ular the renormalized perturbation series (RPS) for it;where S I ( ω ) is defined as usual by G I ( ω ) = (cid:2) ω + − E I − S I ( ω ) (cid:3) − = (cid:2) g − I ( ω ) − S I ( ω ) (cid:3) − (14)with g I ( ω ) = ( ω + −E I ) − the (purely) site-diagonal prop-agator for the extreme MBL limit of T IJ = 0 ( H t ≡ G I (˜ ω ) = µ E G I ( ω ),given from eq. 14 by µ E G I ( ω ) = ˜ G I (˜ ω ) = (cid:104) ˜ ω + − ˜ E I − ˜ S I (˜ ω ) (cid:105) − (15)where ˜ ω + = ˜ ω + i ˜ η and ˜ η = η/µ E ; with the rescaled FSsite-energy (cf. eq. 11)˜ E I = ( E I − E ) /µ E , (16)and where the self-energy in consequence rescales as˜ S I (˜ ω ) = S I ( ω ) /µ E = ˜ X I (˜ ω ) − i ˜∆ I (˜ ω ) . (17)The local propagator ˜ G I (˜ ω ) = (cid:80) n | A nI | / (˜ ω + − ˜ E n )(with ˜ E n := ( E n − E ) /µ E ), so the local DoS˜ D I (˜ ω ) = (cid:88) n | A nI | ˜ η/π (˜ ω − ˜ E n ) + ˜ η ≡ (cid:88) n | A nI | δ (˜ ω − ˜ E n ) , (18) such that the total DoS, ˜ D (˜ ω ) = N H (cid:80) I ˜ D I (˜ ω ), is thestandard normal form eq. 12.In the following we focus directly on the Feenberg self-energy ˜ S I (˜ ω ), and in particular its imaginary part ˜∆ I (˜ ω )(long used in the context of 1BL ), which correspondsphysically to the inverse lifetime for FS site/state I par-ticipating in states of energy ˜ ω . Our analysis rests onthe contention that its behavior discriminates betweenlocalized and extended states, namely that for extendedstates ˜∆ I (˜ ω ) is non-vanishing with probability unity overan ensemble of disorder realizations; while for localizedstates by contrast ˜∆ I (˜ ω ) is vanishingly small with prob-ability one, specifically is proportional to the imaginarypart of the energy (˜ η → y = ˜∆ I / ˜ η itself isfinite. To support this we sketch in Appendix A somesimple arguments indicating, whether 1BL or MBL isconsidered, that for extended states ∆ I ( ω ) is non-zeroand is ∝ µ E in the many-body case, while ∆ I ( ω ) ∝ η isvanishingly small for localized states. We add that thebehavior described is also consistent with our numericalresults.Since ˜∆ I is distributed, it is in effect a probabilisticorder parameter. One should thus study its distribu-tion – for extended states the probability density F ( ˜∆ I )over an ensemble of disorder realizations that any FS sitehas a particular ˜∆ I (˜ ω ) (physically, the distribution of lo-cal inverse lifetimes/rates for FS states I ). Likewise forlocalized states one considers the corresponding density˜ F ( y ) for y = ˜∆ I / ˜ η , given by ˜ F ( y ) = ˜ ηF (˜ ηy ). It is thesedistributions we will consider in the following sections.Before proceeding, we draw attention to two significantpoints. First, note that the disorder averaged (cid:104) ˜∆ I (˜ ω ) (cid:105) – which amounts to Fermi’s golden rule rate in leadingorder (see below) – is as well known always non-zero, re-gardless of whether states are localized or extended (astoo are the higher moments (cid:104) [ ˜∆ I (˜ ω )] p (cid:105) , p ≥ (cid:104) ˜∆ I (˜ ω ) (cid:105) is not an order parameter for the MBL transition(although one anticipates it will be adequately ‘typical’of F ( ˜∆ I ) sufficiently deep in a regime of extended states).This has immediate implications for the distribution ˜ F ( y )of y = ˜∆ I / ˜ η characteristic of the localized regime, be-cause, since ˜ η ≡ its moments – other than p = 0 (normalization) – must in consequence diverge.The latter in turn suggests that the large- y asymptoticbehavior of ˜ F ( y ) may be a power-law, ˜ F ( y ) ∼ y − ξ withthe exponent in the range 1 < ξ <
2. This is indeed thecase, as will be seen in subsequent sections.Second, consider the averaged (cid:104) ˜∆ I (˜ ω ) (cid:105) itself. Fig. 1shows numerical results for (cid:104) ˜∆ I (˜ ω ) (cid:105) vs disorder strength W/t , for the d = 1 case with a standard box site-energydistribution P ( (cid:15) ) = θ ( W − | (cid:15) | ) of full width W ( θ ( x ) isthe unit step function). We consider half-filling ν = 1 / V /t = 2 (corresponding for the ran-dom XXZ model to S z tot = 0 with spin-isotropic Heisen-berg exchange), and states at the band center (˜ ω = 0).With these parameters, a range of previous exact diago-nalization studies estimate the MBL transition to occur0 5 10 15 20 2500 . . . . W/t h e ∆ I i . . ↑ N h ∆ I i FIG. 1. The averaged (cid:104) ˜∆ I (cid:105) vs disorder W/t for band cen-ter states (˜ ω = 0) in d = 1; shown as a function of systemsize for N = 10 (red), 12 (blue), 14 (green), 16 (purple),as obtained by exact diagonalization. Results are for a box P ( (cid:15) ) = θ ( W − | (cid:15) | ), at half-filling, and for V /t = 2. (cid:104) ˜∆ I (cid:105) con-verges rapidly with increasing N , and is well converged by N (cid:39)
12. The Fermi golden rule result (eq. 19) is shown forcomparison (dashed line).
Inset : Corresponding system sizeevolution of (cid:104) ∆ I ( ω ) (cid:105) (= µ E ˜∆ I (˜ ω )). Discussion in text. for W/t in the range ∼ − (for W and t as we have defined them), although a numerical linked-cluster expansion method suggests these may under-estimate somewhat the critical W/t . Results here areobtained by exact diagonalization, for system sizes of N = 10 −
16 real-space sites (further discussion will begiven in sec. VI). As seen clearly in fig. 1, (cid:104) ˜∆ I (˜ ω ) (cid:105) con-verges rapidly as a function of system size N . It is infact already well converged by N (cid:39)
12 for essentiallyall
W/t save the lowest. By contrast, the inset to fig. 1shows the corresponding behavior for (cid:104) ∆ I ( ω ) (cid:105) , which forany given W/t is seen to increase ( ∝ √ N ) with systemsize. The latter is expected, since ∆ I ( ω ) = µ E ˜∆ I (˜ ω )(eq. 17) and µ E ∝ √ N (eqs. 9,8,5). It is also readily un-derstood physically, for (cid:104) ∆ I ( ω ) (cid:105) is given to leading order(in the Feenberg RPS) by the Fermi golden rule result (cid:104) ∆ I ( ω ) (cid:105) (cid:39) πZt D ( ω ), with D ( ω ) ∝ /µ E the GaussianDoS eq. 10. Since the mean coordination number Z ∝ N (eq. 5), (cid:104) ∆ I ( ω ) (cid:105) ∝ √ N ; whence (cid:104) ˜∆ I (˜ ω ) (cid:105) (cid:39) πt Zµ E D ( ω ) = πt Zµ E ˜ D (˜ ω ) (19)(with ˜ D (˜ ω ) the standard normal DoS eq. 12), and indeedremains finite as the thermodynamic limit N → ∞ is ap-proached. As seen in fig. 1 (dashed line), the golden ruleresult eq. 19 captures rather well the evolution of (cid:104) ˜∆ I (˜ ω ) (cid:105) with disorder (and in fact becomes quantitatively accu-rate with increasing disorder).Although thus far illustrated primarily for the aver-age (cid:104) ˜∆ I (˜ ω ) (cid:105) , the latter considerations underscore the im-portance of the energy rescaling eq. 11, and in particu- lar the consequent rescaling (eq. 17) of the self-energy,˜∆ I (˜ ω ) = ∆ I ( ω ) /µ E (see also Appendix A for a furtherargument for this): it is ˜∆ I (˜ ω ) which in the thermody-namic limit is non-vanishing and finite with probabilityunity in an extended regime. It is thus its probability dis-tribution F ( ˜∆ I ) which must be considered, rather thanthat for ∆ I . IV. A MEAN-FIELD APPROACH
Here and in sec. V we develop a self-consistent mean-field approach to MBL which, while undoubtedly sim-ple, appears to capture at least some key features of theproblem. The approach has its antecedents in a ‘typicalmedium’ theory introduced many years ago, in workon localization and vibrational energy flow in a many-body quantum state space, following a similar earlier ap-proach to one-body localization. As mentioned above we consider the Feenberg self-energy ˜ S I for the local FS propagator ˜ G I , in particu-lar the renormalized perturbation series (RPS) for it (which follows from analysis of eq. 13). With this, theself-energy is expressed as a function of the local propa-gators { ˜ G J } , such that determining ˜ S I becomes a ques-tion of self-consistency (in a probabilistic sense). For themany-body system of interest the RPS has exactly thesame structure as for a one-body TBM, because the Fock-space Hamiltonian eq. 2 is of TBM form. The RPS for S I ( ω ) is given as a sum of all closed, self-avoiding hop-ping paths on the FS lattice which begin and end on FSsite I , contain n ≥ t (with onlyeven n possible since the FS lattice is bipartite ), andwhere intermediate vertices corresponding to sites J (cid:54) = I contain the local propagator for that site. Specifically S I ( ω ) = (cid:88) J T IJ G J ( ω ) + .... = t (cid:88) J ω + − E J − S J ( ω ) + .... (20)with the second-order ( n = 2) term shown explicitly,the sum being over the Z I Fock-space sites J whichare connected to site I under the hopping t ; and with+ ... referring to higher-order ( n ≥
4) RPS terms. With˜ G I = µ E G I (eq. 15), ˜ S I = S I /µ E (eq. 17) follows as ˜ S I (˜ ω ) = t µ E (cid:88) J ˜ G J (˜ ω ) = t µ E (cid:88) J ω + − ˜ E J − ˜ S J (˜ ω )(21)We have dropped the higher-order RPS terms here, be-cause in practice we consider the problem only at second-order level. This is well known to be exact if the Fock-space lattice has the topology of a Bethe lattice, which it does not. Here we employ it as a natural leading-order approximation, conjecturing that higher-order RPScontributions will produce only a quantitative modifica-tion of results arising at second-order level (for whichthe numerical results of sec. VI, which do not make thisapproximation, provide support).Granted this, we then analyze the problem simply andapproximately in the spirit of a probabilistic mean-fieldtheory, via the following strategy:(a) The self-energy ˜ S J in eq. 21 for sites J connected to I is first replaced by a typical value, denoted ˜ S t (˜ ω ) =˜ X t (˜ ω ) − i ˜∆ t (˜ ω ).(b) With this, the distribution F ( ˜∆ I ) of ˜∆ I is obtained.(c) Self-consistency is then imposed by requiring that atypical value of ˜∆ I arising from this distribution coin-cides with the input ˜∆ t (˜ ω ). In practice, ‘typicality’ ismeasured by the geometric mean,ln ˜∆ t (˜ ω ) = (cid:104) ln ˜∆ I (˜ ω ) (cid:105) (22)with the average (cid:104) ... (cid:105) = (cid:82) d ˜ ω ...F ( ˜∆ I ) over the distri-bution F ( ˜∆ I ). ˜∆ t (˜ ω ) acts in effect as an order pa-rameter, being non-zero if states of energy ˜ ω are ex-tended; and vanishingly small in the MBL phase, with˜∆ t (˜ ω ) ∝ ˜ η → S I (˜ ω ) = t µ E (cid:88) J ω + − ˜ E J + i ˜∆ t (˜ ω ) (23)where for convenience we have subsumed Re ˜ S t (˜ ω ) =˜ X t (˜ ω ) into the energy, by defining ω ≡ ω (˜ ω ) as ω = ˜ ω − ˜ X t (˜ ω ) . (24)Now consider the Fock-space site-energy E J for anysite J connected to I under hopping. Since the hoppingis between NN real-space sites, E J differs from E I simplyby the difference in site-energies of sites between whichthe fermion hops, and by the resultant change in the NNinteraction contribution; e.g. for d = 1, E J = E I + (cid:15) i +1 − (cid:15) i + V [ n i +2 − n i − ]with (cid:15) i +1 the site-energy of the occupied real-space sitein J ( ≡ | J (cid:105) ) to which a fermion hops under T IJ , and (cid:15) i that for the NN real-space site occupied in I from whenceit came [the occupation numbers n i +2 and n i − are thesame for both I and J ]. The E J are thus highly correlatedwith E I , and differ from it by an amount of O ( W, V ), with W the disorder scale on which the real-space site-energiesfluctuate. Hence ˜ E J = E J /µ E entering eq. 23 is˜ E J = ˜ E I + O (cid:16) Wµ E , Vµ E (cid:17) ≡ ˜ E I , (25)since µ E ∝ √ N . The key point here is that the rescaledFS site-energies ˜ E J entering eq. 23 for ˜ S I (˜ ω ) are the sameas that for site I in the thermodynamic limit, i.e. areeffectively resonant with it.Eq. 23 thus reduces to˜ S I (˜ ω ) = t µ E (cid:88) J ω + − ˜ E I + i ˜∆ t (˜ ω ) (26a)= Z I t µ E ω + − ˜ E I + i ˜∆ t (˜ ω ) . (26b) All terms in the sum in eq. 26a are the same, and theirnumber is the coordination number Z I of site I . Thishas a mean of Z ∝ N (eq. 5) and a standard deviation µ Z ∝ √ N . Since µ E ∝ N , only the mean Z is relevantin the thermodynamic limit, so we replace Z I ≡ Z in eq.26b. ˜∆ I (˜ ω ) = − Im ˜ S I (˜ ω ) then follows as˜∆ I (˜ ω ) = Γ [˜ η + ˜∆ t (˜ ω )][ ω − ˜ E I ] + [˜ η + ˜∆ t (˜ ω )] , (27)where Γ = Zt /µ E is thus defined and is finite as N →∞ , being given by (eqs. 5,8,9)Γ = Zt µ E = t (cid:2) t + d (cid:104) (cid:15) (cid:105) + ν (1 − ν ) V (cid:3) ≤ . (28)Eq. 27 will be used in the following section to determinethe probability distribution of ˜∆ I (˜ ω ). Before that it ishowever useful to have some insight into (a) the physicalorigin of the factor Z I t /µ E in the basic expression eq.26b for ˜ S I ; and (b) some key differences between thepresent problem and that arising in 1-body localization.
1. Physical digression
There are two distinct elements contributing to thefactor of Z I t /µ E in eq. 26. First, an effective rescalingof the hopping, t → t/µ E . This is a general consequenceof rescaling the energy as in eqs. 11,16, and hence thelocal propagator as ˜ G I = µ E G I (eq. 15). To see thisdirectly define ˜ G IJ = µ E G IJ generally, so the ‘equationof motion’ eq. 13 reads(˜ ω + − ˜ E I ) ˜ G IJ (˜ ω ) = δ IJ + (cid:88) K ˜ T IK ˜ G KJ (˜ ω )with ˜ T IK = T IK /µ E (and | ˜ T IK | = t/µ E ). Comparisonto eq. 13 shows that the local ˜ G I in particular, and hence˜ S I , are the same functions of { ˜ T JK } (and { (˜ ω − ˜ E J ) } ) that G I and S I are of T IK (and { ( ω − E J ) } ); or equivalentlyfor the RPS itself, that ˜ S I is the same function of t/µ E and the { ˜ G J } that S I is of t and the { G J } . This effec-tive rescaling of t → t/µ E ∝ t/ √ N is analogous to thatrequired in dynamical mean-field theory to ensure thatthe limit of infinite spatial dimensions is well-defined.To illustrate the physical origin of the coordinationnumber factor in Z I t /µ E , consider a particularly simplelimit: of ‘one shell’, where a given FS site I is coupledunder the hopping H t to its Z I ∝ N neighbors { J } , them-selves uncoupled from each other (eq. 26 above, obviouslywith ˜ S t = 0, in fact captures this limit exactly). This isformally equivalent to a non-interacting Anderson impu-rity coupled to a ‘conduction band’ containing Z I ∝ N states (the ‘impurity’ being I , band states the { J } ). Inthis case the hopping contribution H t is given by H t = | I (cid:105) (cid:88) J T IJ (cid:104) J | + h . c ., (29)showing that | I (cid:105) couples directly only to the particularlinear combination of states denoted by | (cid:105) ∝ (cid:80) J T IJ | J (cid:105) .This corresponds to the so-called 0-orbital in the Wil-son chain representation of the Anderson model, and isgiven in normalized form by | (cid:105) = 1 √ Z I t (cid:88) J T IJ | J (cid:105) (since (cid:80) J T IJ = Z I t ). In terms of it, H t (eq. 29) is thus H t = (cid:112) Z I t | I (cid:105)(cid:104) | + h . c . (30)so that | I (cid:105) and | (cid:105) are coupled by an effective hop-ping of √ Z I t . Combined with the t → t/µ E rescal-ing above, the effective hopping entering ˜ S I (˜ ω ) is thus t eff = √ Z I t/µ E (the square of which naturally appearsin the second-order RPS expression eq. 26b, or eq. 27);with t eff , and hence Γ = Zt /µ E (eq. 28), thus remainingfinite as N → ∞ .It may also be helpful to contrast the situation arisingabove for MBL with that occurring for 1-body localiza-tion. In the latter case the counterpart of eq. 20 for theself-energy S i ( ω ) of real-space site i is S i ( ω ) = t (cid:88) j ω + − (cid:15) j − S j ( ω )with the sum over the NN sites to i (of which there are,say, K c ). Since the site-energies { (cid:15) k } are independentrandom variables, the right hand side of this expressionis a sum of K c independent random terms; in contraste.g. to eq. 26a for MBL where, since the Z I terms inthe sum coincide, there is only a single random term.In addition, the probability distribution for S i ( ω ) in thiscase is independent of the site energy of site i ; while inthe MBL case the distribution of ˜ S I both depends on theFock-space site-energy ˜ E I for that site and, at the levelof eq. 26, is in fact entirely determined by it. V. SELF-CONSISTENT DISTRIBUTIONS ANDMBL TRANSITION
Using eq. 27 for ˜∆ I , its distribution (for any givenenergy ˜ ω ) is obtained by integrating over the distribution P ( ˜ E I ) of ˜ E I = ( E I − E ) /µ E (eq. 16), F ( ˜∆ I ) = (cid:90) ∞−∞ d ˜ E I P ( ˜ E I ) δ (cid:16) ˜∆ I − Γ [˜ η + ˜∆ t (˜ ω )][ ˜ E I − ω ] + [˜ η + ˜∆ t (˜ ω )] (cid:17) . (31)From sec. II A, P ( ˜ E I ) is a Gaussian with vanishing mean, P ( ˜ E I ) = 1 √ πλ exp (cid:32) − ˜ E I λ (cid:33) , (32) with λ = µ E /µ E independent of N and given (from eqs.8,9,5) by λ = µ E µ E = (cid:104) (cid:15) (cid:105) + ν (1 − ν ) dV (cid:104) (cid:15) (cid:105) + ν (1 − ν ) dV + 2 dt . (33)From now on in this section we focus explicitly on theband center ˜ ω = 0, eqs. 11,12 (equivalently ω = 0, eq.24, since ˜ X t (˜ ω = 0) vanishes by symmetry). With theform eq. 32 for P ( ˜ E I ), eq. 31 is readily evaluated to give F ( ˜∆ I ) =1 (cid:113) − ˜∆ I [˜ η + ˜∆ t ]Γ (cid:114) κπ I exp (cid:104) − κ (cid:16) I − [˜ η + ˜∆ t ]Γ (cid:17)(cid:105) (34)for 0 ≤ ˜∆ I < Γ / (˜ η + ˜∆ t ) (and zero otherwise), where κ = Γ(˜ η + ˜∆ t )2 λ (35)is thus defined. Eq. 34 encompasses both the MBL anddelocalized regimes. We now consider them separately. A. MBL regime
In the MBL regime y = ˜∆ I / ˜ η is finite, so one considersits distribution ˜ F ( y ) = ˜ ηF (˜ ηy ) (sec. III). Since ˜ η = 0+,eq. 34 then gives˜ F ( y ) = (cid:114) ˜ κπ y exp (cid:16) − ˜ κy (cid:17) : y = ˜∆ I ˜ η (36)(holding for all y ∈ (0 , ∞ )), where (eq. 35)˜ κ = κ ˜ η = Γ2 λ (cid:16) t ˜ η (cid:17) . (37)Eq. 37 is precisely a L´evy distribution, with a characteris-tic long tail ∝ y − / , such that all moments of y diverge.For the reasons explained in sec. III, this is a physicallynatural form for the distribution of ˜∆ I / ˜ η in the MBLphase. We will compare this behavior to results obtainedby exact diagonalization in sec. VI (where none of the ap-proximations entering the mean-field theory are made).Next we impose self-consistency on the geometric mean(eq. 22). From the normalized ˜ F ( y ) eq. 36, the average (cid:104) ln( ˜∆ I / ˜ η ) (cid:105) = (cid:82) ∞ dy ˜ F ( y ) ln y follows, (cid:68) ln (cid:16) ˜∆ I ˜ η (cid:17)(cid:69) = ln ˜ κ − √ π (cid:90) ∞ dx ln x e − x (38a)= ln(4˜ κ ) + γ : γ = 0 . .. (38b)with γ the Euler-Mascheroni constant. Imposingln( ˜∆ t / ˜ η ) = (cid:104) ln( ˜∆ I / ˜ η ) (cid:105) (eq. 22) gives ˜∆ t / ˜ η = 4 e γ ˜ κ , with˜ κ given by eq. 37. Hence the self-consistency condition y t = [2 e γ Γ /λ ](1 + y t ) for y t = ˜∆ t / ˜ η ( ≥ t ˜ η = T (cid:2) − T (cid:3) − : T ≤ T ( ≥
0) defined by T = 2 e γ Γ λ = 2 e γ Zt ( λµ E ) . (40)Note that Γ (eq. 28) and λ (eq. 33) are finite as N → ∞ (with Z and µ E each ∝ N ), whence T remains boundedin the thermodynamic limit. Since ˜∆ I / ˜ η is non-negativeand finite with probability one throughout the MBLphase, eq. 39 shows that the phase is self-consistent onlyfor T <
1. The transition to delocalization from the MBLphase thus occurs as T → − , where ˜∆ t / ˜ η ∼ [1 − T ] − s diverges with an exponent of unity (fig. 2).The mean-field transition criterion T = 1 will beconsidered further below. Here we simply note that Γ(eq. 28), λ (eq. 33), and hence T , are invariant under ν ↔ (1 − ν ). Filling fractions ν (= N e /N ) and (1 − ν )are thus equivalent; as required physically, and reflectingthe invariance of H under a particle-hole transformation.The self-consistent ˜ κ entering the L´evy distribution eq.36 follows from eqs. 37,39,40 as˜ κ = 14 e γ ˜∆ t ˜ η = 14 e γ T − T . (41)It sets the scale for the emergence of the ∼ y − / tailscharacteristic of the distribution, which arise (eq. 36) for y (cid:29) ˜ κ , and get pushed to progressively larger y -valueson approaching the transition T → − .More importantly, note that the L´evy distribution for y = ˜∆ I / ˜ η can be written in the one-parameter scalingform˜ F ( y ) = 1 c ˜ κ f L (cid:16) yc ˜ κ (cid:17) = 1˜ cκ f L ( x ) (42a) f L ( x ) = 1 √ πc x exp (cid:0) − cx (cid:1) : x = yc ˜ κ , (42b)with c an arbitrary constant (independent of the physicalparameters); and with f L ( x ) – the probability density of x = y/ ( c ˜ κ ) – dependent solely on x . If e.g. c = 4 e γ ischosen (with γ again Euler’s constant), then since thegeometric mean of eq. 36 is y t = 4 e γ ˜ κ , eq. 42 reads˜ F ( y ) = y − f L ( x ) with f L ( x ) the probability density of x ≡ y/y t . The central point here is that all L´evy distribu-tions can be scaled onto each other, with f L ( x ) as suchcharacteristic of the entire MBL phase. We return tothis important point when considering numerical resultsin sec. VI.
B. Delocalized regime
In a delocalized regime ˜∆ I (and hence ˜∆ t ) is finite,and since ˜ η = 0+ eq. 34 for F ( ˜∆ I ) thus reduces to F ( ˜∆ I ) = (cid:114) κπ I exp (cid:16) − κ ˜∆ I (cid:17) × (cid:113) − ˜∆ I ˜∆ t Γ exp (cid:16) κ ˜∆ t Γ (cid:17) (43)with κ = Γ ˜∆ t / (2 λ ). This is the product of a L´evydistribution (in ˜∆ I itself), times a contribution that isintegrably divergent as the upper limit ˜∆ I = Γ / ˜∆ t of thedistribution is approached (that limit acting as a cutoffto the L´evy tail).To determine ln ˜∆ t = (cid:104) ln ˜∆ I (cid:105) self-consistently it ismore economical to work directly with eqs. 31,32 for F ( ˜∆ I ); from which (with ˜ η = 0 and ω = 0), (cid:104) ln ˜∆ I (cid:105) = 1 √ πλ (cid:90) ∞−∞ dx e − x λ ln (cid:104) Γ ˜∆ t x + ˜∆ (cid:105) (44a)= 2 √ π (cid:90) ∞ dy e − y ln Γ ˜∆ t λ y (cid:16) ˜∆ λ y (cid:17) (44b)Hence (noting eqs. 38)ln ˜∆ t = ln (cid:20) e γ Γ λ ˜∆ t (cid:21) − √ π (cid:90) ∞ dy exp( − y ) ln (cid:32) λ y (cid:33) (45)where the first term is recognized (eq. 40) as beingln( T ˜∆ t ). The low- ˜∆ t behavior of the integral in eq. 45 is (cid:90) ∞ dy exp( − y ) ln (cid:16) xy (cid:17) x → ∼ π √ x − √ πx + O (cid:0) x (cid:1) . Eq. 45 thus yields the self-consistency condition for ˜∆ t in the delocalized phase close to the transition,˜∆ t ˜∆ t → ∼ T ˜∆ t (cid:16) − √ πλ ˜∆ t + [1 + π ] λ ˜∆ + O ( ˜∆ ) (cid:17) . Since ˜∆ t ≥ T ≥
1, given to leading order by˜∆ t T → ∼ λ √ πT [ T − s : s = 1 . (46)The transition approached from the delocalized phasethus occurs (as it ought) at the same point T = 1 asthe approach to it from the MBL phase; and ˜∆ t vanishesas T →
1+ with the same exponent, s = 1, with which˜∆ t / ˜ η diverges as the transition is approached from theMBL side (eq. 39).Throughout the delocalized phase more generally, theself-consistent ˜∆ t is obtained numerically. One further0 5 15 20 250123 Ergodic MBL W/t W crit /t a · Wtba · Vt FIG. 2. Evolution with disorder (
W/t ) of the self-consistentmean-field ˜∆ t (delocalized/ergodic phase) and ˜∆ t / ˜ η (MBLphase), for the band center ˜ ω = 0. Shown for a box site-energy distribution, for d = 1 at half-filling with V /t = 2.The dashed line indicates the transition point.
Inset : Mean-field phase diagram in the (
W/t, V /t )-plane (with constants a, b given by eq. 47). Discussion in text. limit is however readily extracted. As the width, λ , ofthe distribution of ˜ E I (eq. 32) vanishes, F ( ˜∆ I ) → δ ( ˜∆ I − ˜∆ t ) tends to a δ -distribution. In this case eq. 44a givesln ˜∆ t = (cid:104) ln ˜∆ I (cid:105) = ln(Γ / ˜∆ t ), and hence˜∆ t λ → ∼ √ Γ = t (cid:113) t + d (cid:104) (cid:15) (cid:105) + ν (1 − ν ) V ∼ λ limit of theFermi golden rule behavior (see eq. 19). The behavior considered above and in sec. V A isexemplified in fig. 2, showing the evolution of ˜∆ t and ˜∆ t / ˜ η (for the MBL phase) with disorder, W/t .Results are shown for the box site-energy distribution P ( (cid:15) ) = θ ( W − | (cid:15) | ) (for which (cid:104) (cid:15) (cid:105) = W / d = 1 at half-filling with interaction V /t = 2.Now consider further the simple mean-field transitioncriterion, viz. T = 1 with T given by eq. 40. From eqs.40,33,28 (with (cid:104) (cid:15) (cid:105) = W /
12 as above), T = a b (cid:0) VW (cid:1) (cid:104) tW (cid:105) : a =4 e γ/ √ d (cid:39) . √ d, b = 2 (cid:112) ν (1 − ν ) d (47)Defining x = W/at and y = bV /at , the condition T < y > − x . The resultant phaseboundary is shown in fig. 2 (inset), and the followingpoints should be noted. (1) A transition arises for allspace dimension d , including for the non-interacting limit V = 0. The latter is of course wrong for d = 1 ,
2; but isas expected from a mean-field theory, which may handleadequately generic behavior above a lower critical dimen-sion but not below it. (2) As seen from the V -dependence 0 0 . . . . . e ∆ I F ( e ∆ I ) W/t = 3
W/t = 5
W/t = 71
FIG. 3. Mean-field F ( ˜∆ I ) vs ˜∆ I (eq. 43) in the delocal-ized phase; for the same parameters as figs. 1,2 and shown for W/t = 3 , , W crit /t (cid:39) . F ( ˜∆ I )emerges (shown for the W/t = 7 case, dashed line). Discus-sion in text. of the phase boundary in fig. 2 inset, increasing the in-teraction for given, sufficiently low disorder, eventuallydrives the system to an MBL phase, as one anticipatesphysically (interactions effectively self-generate disorderin the distribution of {E I } ). This is indeed as found byexact diagonalization (although a transition occurringat a finite- V /t as the disorder vanishes, rather than e.g.as
V /t → ∞ , is presumably an artifact of the theory). (3)The transition for d = 1 at half-filling and for V /t = 2occurs at
W/t (cid:39) .
6. As mentioned earlier, a range ofexact diagonalization studies for these parameters esti-mate the MBL transition to occur for
W/t in the range ∼ − so the mean-field estimate appearsnot wildly out of line.Finally here, we comment briefly on the mean-fielddistribution F ( ˜∆ I ) in the delocalized phase. From eq.43, this is the product of a L´evy distribution (in ˜∆ I it-self), times a factor that is integrably divergent as theupper limit ˜∆ I = Γ / ˜∆ t of the distribution is approached(and which in passing we add can be shown to generate F ( ˜∆ I ) → δ ( ˜∆ I − ˜∆ t ) as λ →
0, mentioned above). Herewe simply note from eq. 43 that the L´evy contributionitself (whose mode occurs at ˜∆ I = 2 κ/
3) is ‘well formed’in an obvious sense if ˜∆ I (cid:29) κ = (Γ / λ ) ˜∆ t ∝ ˜∆ t , whilethe factor [1 − ˜∆ I ( ˜∆ t / Γ)] − / is unity for practical pur-poses provided ˜∆ I (cid:28) Γ / ˜∆ t ∝ / ˜∆ t . On approachingthe transition where ˜∆ t →
0, one thus expects to seethe emergence of L´evy behavior in F ( ˜∆ I ) over an in-creasingly wide range of ˜∆ I ; as illustrated in fig. 3, fromwhich L´evy-like behavior is seen to emerge reasonablyfar into the delocalized phase. We return to this whenconsidering exact diagonalization results in sec. VI B.0
1. Energy dependence
While we have focused above on the center of theeigenvalue spectrum, ˜ ω = ( ω − E ) /µ E = 0, the anal-ysis can naturally be extended to ˜ ω (cid:54) = 0. We com-ment on it briefly. In this case the real part ˜ X t (˜ ω ) =Re ˜ S t (˜ ω ) no longer vanishes by symmetry, and is re-lated to ˜∆ t (˜ ω ) = − Im ˜ S t (˜ ω ) by the Hilbert transform π ˜ S t (˜ ω ) = (cid:82) ∞−∞ d ˜ ω (cid:48) ˜∆ t (˜ ω (cid:48) ) / (˜ ω + − ˜ ω (cid:48) ); thus determiningthe ˜ ω -dependence of ω = ˜ ω − ˜ X t (˜ ω ) (eq. 24), and henceof the general probability density eq. 31 for F ( ˜∆ I ) ≡ F ( ˜∆ I ; ˜ ω ). Results arising mirror those obtained abovefor ˜ ω = 0. In the MBL phase for example, the self-consistent ˜∆ t (˜ ω ) / ˜ η has the same form as 39, viz.˜∆ t (˜ ω )˜ η = T (˜ ω )[1 − T (˜ ω )] − : T (˜ ω ) ≤ T (˜ ω ) < T (˜ ω ) = T ( − ˜ ω ). The transition for energy ˜ ω thusoccurs as T (˜ ω ) → − . This condition determines themobility edges, ˜ ω ± = ± ˜ ω m , separating regions of local-ized and delocalized states. T (˜ ω ) is readily shown to beof form T (˜ ω ) = T (0) exp (cid:0) − I (˜ ω ) (cid:1) (49)with T (0) = 2 e γ Γ /λ the band center result (eq. 40), I (˜ ω ) ˜ ω → ∼ α ˜ ω + O (˜ ω ) (with α > O (1));and with I (˜ ω ) = I ( − ˜ ω ) more generally an increasingfunction of ˜ ω from the band center ˜ ω = 0, whence T (˜ ω )decreases with ˜ ω . Band center states are thus the lastto become MBL with increasing disorder, and resultantmobility-edge trajectories in the ( W/t, ˜ ω )-plane accord-ingly have the expected characteristic ‘D-shape’. Notethat since the mobility edges open up continuously ondecreasing disorder W below the critical value for bandcenter delocalization, they thus occur at finite valuesof ˜ ω m ≡ ( ω m − E ) /µ E ; i.e. for ω m − E ∝ √ N (since µ E ∝ √ N ), as pointed out on general grounds in [59]. VI. NUMERICAL RESULTS
Results obtained by exact diagonalization (ED) arenow considered, for a d = 1 open chain with site-energydistribution P ( (cid:15) ) = θ ( W − | (cid:15) | ). While other parameterregimes have been studied, here we consider explicitlyhalf-filling ( ν = 1 /
2) with interaction
V /t = 2, and forstates at the band center ˜ ω = 0. ; for which the MBLtransition occurs for W/t ∼ − Since our main aim is to determine the probability dis-tributions of the ˜∆ I (or ˜∆ I / ˜ η ), and their evolution withdisorder W/t , we first describe how these are calculatedfor finite-size systems. The self-energy ˜ S I (˜ ω ) – in itsentirety (rather than e.g. at truncated RPS level) – is defined via the inverse of the local propagator ˜ G I (˜ ω ), eq.15; from which ˜∆ I (˜ ω ) = − Im ˜ S I (˜ ω ) follows,˜∆ I (˜ ω ) = Im (cid:16) G I (˜ ω ) (cid:17) − ˜ η ˜ G I (˜ ω ) = (cid:88) n | A nI | ˜ ω + i ˜ η − ˜ E n (50)(where ˜ η = η/µ E ). To obtain ˜∆ I (˜ ω ), in principle oneconsiders first the thermodynamic limit N → ∞ , fol-lowed by η → A nI (eq. 4)are obtained from ED – the thermodynamic limit ob-viously cannot be taken, whence in turn η cannot beset to zero from the outset. This just reflects the factthat any finite-size system, no matter how large, strictlyspeaking has a discrete eigenvalue spectrum. The small-est typical energy scale in that spectrum is howeverthe eigenvalue spacing [ N H D ( ω )] − , and it is an η ofthis order that should be considered in finite- N cal-culations (which amounts simply to regularizing the δ -functions in e.g. D ( ω ) = N − H (cid:80) n δ ( ω − E n ), replac-ing them by Lorentzians of halfwidth η ). We thus take η = [ N H D ( ω )] − , and hence ˜ η = [ N H ˜ D (˜ ω )] − with ˜ D (˜ ω )the standard normal DoS eq. 12, i.e.˜ η = 1 N H ˜ D (˜ ω ) ˜ ω =0 = √ πN H (51)such that ˜ η ∝ /N H ∼ e − cN is exponentially small inthe number of sites N . We have taken the √ π prefactorindicated in eq. 51 (but have naturally confirmed thatresults are insensitive to this choice).With the procedure sketched, the distributions are de-termined by averaging over disorder realizations and FSsites; with F ( ˜∆ I ) the probability density, over an ensem-ble of disorder realizations, that any site has a particularvalue of ˜∆ I . Calculations are for system sizes N = 10 − N H ∼ − N = 16.These distributions per se will be considered in thefollowing sections. But first (fig. 4) we give an overviewof the resultant geometric mean ˜∆ t of F ( ˜∆ I ), and itsevolution with disorder W/t and system size N . As arelevant comparator, the inset to fig. 4 shows a mea-sure often used to distinguish localized from delocalizedstates; viz. the ratio of consecutive eigenvaluespacings, r n = min( δ n , δ n − ) / max( δ n , δ n − ) where δ n = E n +1 − E n (with E n the ordered eigenvalues). The meanratio r vs W/t is shown, for states in the immediate vicin-ity of the band center and N = 12 −
16. For the Gaussianorthogonal ensemble (GOE) characteristic of extendedstates r GOE = 0 . .. , while for Poissonian statistics ap-propriate to the MBL phase, r Pois = 0 . .. . In theextended regime sufficiently below the critical disorder, r increases with increasing N for given W/t (and for lowenough disorder has effectively reached the GOE limit).10 5 10 15 20 2500 . . . W/t . . . . GOEPoisson r FIG. 4. Geometric mean ˜∆ t (circles) of F ( ˜∆ I ) vs disorder W/t , for system sizes N = 12 (blue), 14 (green), 16 (purple).Arithmetic means (cid:104) ˜∆ I (cid:105) (crosses) for the same sizes are alsoshown (as in fig. 1), and are well converged in N by N (cid:39) Inset : Mean ratio r (squares) of consecutive eigenvaluespacings vs W/t for the same system sizes. Discussion in text.
In the MBL regime by contrast r shows the reverse trend,decreasing with increasing N towards the Poisson limit.In between is a continuous crossover, as expected forfinite-size systems. The data are clearly consistent withthe occurrence of the MBL transition (whose existence isnot in doubt ), though to gauge the critical W/t withsome confidence requires larger system sizes coupled witha finite-size scaling analysis. This has been done forsystem sizes up to an impressive N = 22, leading to anestimate of the critical W/t (cid:39) . W/t - and N -dependence of the geometric mean˜∆ t = exp( (cid:104) ln ˜∆ I (cid:105) ); together with those for the arith-metic mean (cid:104) ˜∆ I (cid:105) (= (cid:82) d ˜∆ I ˜∆ I F ( ˜∆ I )) discussed in sec.III (fig. 1). The same qualitative characteristics are seenfor ˜∆ t as for r above: for low enough disorder ˜∆ t in-creases with increasing system size and ultimately tendsto a finite limit (a geometric mean cannot exceed itsarithmetic counterpart); while for larger W/t in the MBLphase the reverse behavior is seen, and it indeed appearslikely that ˜∆ t asymptotically vanishes with increasing N ,as required for localized states. In between is once againan expected crossover behavior (which prevents us beingcredibly quantitative about the critical value of W/t ).Overall, as for r , the data for ˜∆ t are likewise consistentwith the occurrence of the MBL transition. Its behavioris however in marked contrast to that for the arithmeticmean (cid:104) ˜∆ I (cid:105) which, as discussed in sec. III, is well con-verged in N even by N = 12; and, being non-zero inboth the ergodic and MBL phases, does not discriminatebetween them.We turn now to the numerically determined distribu-tions of ˜∆ I and ˜∆ I / ˜ η , beginning with the MBL phase. 10 − − − − − y e F ( y ) N = 10 N = 12 N = 14 N = 161 FIG. 5. Numerical results in the MBL phase for
W/t = 20,showing the distribution ˜ F ( y ) vs y (= ˜∆ I / ˜ η ), for N = 10(red), 12 (blue), 14 (green) and 16 (purple). The dashed lineis a fit of the N = 16 results to a (full) L´evy distribution. A. MBL regime
Fig. 5 shows the distribution ˜ F ( y ) of y = ˜∆ I / ˜ η , forfixed disorder strength W/t = 20 and system sizes N =10 −
16. For each N , a window of power-law tail behavior ∝ y − / occurs; illustrated e.g. by a fit for the N = 16data to a full L´evy distribution, which (eq. 36) has tails ∝ y − / (and captures the numerics rather well downto y ∼ N -dependent cutoff y ∼ O ( N H ) (i.e. ˜∆ I = ˜ ηy ∼ O (1)since ˜ η ∝ /N H , eq. 51); beyond which the distributionfalls of exponentially (itself considered in sec. VI A 1).With increasing N the power-law tails extend over anincreasingly large y -range, which suggests ˜ F ( y ) ∝ y − / as the leading large- y asymptotic behavior of ˜ F ( y ) inthe thermodynamic limit N → ∞ , as arises from themean-field approach of secs. IV,V. To investigate this weconsider a one-parameter scaling ansatz of form˜ F ( y ) = 1 α f (cid:16) yα (cid:17) , (52)with f ( x = y/α ) independent of system size (such that allthe N -dependence resides in α ); and which scaling formalso arises for a pure L´evy distribution (see eq. 42). Withthis ansatz the geometric mean y t follows as y t = αx t ,where x t = exp( (cid:82) ∞ dx f ( x ) ln x ) is N -independent. α is thus proportional to y t ; and we choose α = y t , so f ( x = y/α ) is the distribution of x ≡ y/y t .The resultant f ( x ) is shown in fig. 6, again for W/t =20. f ( x ) now has x − / tails that are common for each N ,and which encompass an increasingly large x -range withincreasing N . The limiting distribution can be estimatedby extrapolating the N = 16 data as an x − / tail (andre-normalizing the distribution with the extrapolated tailin place). This is shown as the dashed line in fig. 6.The range of validity of this distribution is not moreover210 − − − − − − x f ( x ) N = 10 N = 12 N = 14 N = 161 FIG. 6. The distribution f ( x ) vs x of x = y/y t for W/t = 20,and N = 10 (red), 12 (blue), 14 (green) and 16 (purple). Thedashed line is the estimated large- N limiting form of f ( x ). − − − − x f ( x ) − x FIG. 7. f ( x ) vs x = y/y t shown for W/t = 24 (left panel),and 28 (right panel); again for N = 10 (red), 12 (blue), 14(green) and 16 (purple). The dashed line is the estimatedlimiting form of f ( x ) obtained from the W/t = 20 results offig. 6. Results shown are barely distinguishable from thosefor
W/t = 20 in fig. 6. See text for discussion. confined to the power-law tail region, but extends downto x (cid:39) − ; since for the system sizes shown in fig. 6the f ( x ) distributions are quite well converged in N for x (cid:38) − , which comprises the great majority ( ∼ f ( x ) (although is notconverged in N for x (cid:46) − ).The example above refers specifically to W/t = 20,but the same analysis can obviously be performed for
W/t throughout the MBL phase. Results arising are il-lustrated in fig. 7, which in direct parallel to fig. 6 showsthe f ( x ) determined for W/t = 24 and 28; and includesalso (dashed line) the estimated limiting f ( x ) distribu-tion obtained from the W/t = 20 results of fig. 6. Asevident from figs. 7,6, the f ( x ) for these different W/t are scarcely distinguishable (as we have confirmed holdsover a wide range of
W/t , down to
W/t (cid:39) f ( x ) appears to be the same in all cases,attesting to its universality as a function characteristic ofthe entire MBL phase (as discussed in sec. V A, eq. 42,in regard to the pure L´evy distribution arising within themean-field approach).This universality of f ( x ) means (see eq. 52) that the entire y = ˜∆ I / ˜ η -dependence of the distribution ˜ F ( y ) isencoded solely in a single quantity – its geometric mean α = y t , which thus contains all system size and W/t de-pendence. We have naturally investigated this, and findthe following qualitative behavior. For any given systemsize N , increasing disorder W/t and thus moving furtherinto the MBL phase leads to a progressive decrease in y t .This is just as expected physically.However on fixing W/t and progressively increasing thesystem size, α = y t does not appear to saturate (whichthe considerations of sec. V A imply it should); but growsprogressively with N for the system sizes up to N = 16that can realistically be studied [ ˜∆ t and ˜ η ∝ /N H eachdecrease with increasing N in the MBL phase, fig. 4, but y t = ˜∆ t / ˜ η itself increases over the accessible N -range].The obvious question is why, and a plausible explana-tion would seem to be that the system sizes accessiblein practice are not large enough to establish the conver-gence of y t . But it is clearly desirable to have evidence forthat. To this end we now consider the case of one-bodylocalization (1BL), which in turn provides some furtherinsight into the behavior of the MBL distributions shownin figs. 5-7.
1. 1-body localization and back to MBL
It is natural to ask the same questions about 1BL( N e = 1, ν = 1 /N ), for in that case one can easily con-sider much larger system sizes N than for MBL.For d =1, where all states are localized for any disorder,we calculate the distribution ˜ F ( y ) of y = ∆ I /η ( ≡ ˜∆ I / ˜ η )in direct parallel to the procedure above for the MBLcase. With η = [16 t + W ] / /N (:= µ E /N with µ E here simply defined), the ∆ I /η are obtained via eq. 50for states in the immediate vicinity of the band center.The scaling form eq. 52 holds in this case (as detailedbelow), and the value of N for which the geometric mean α = y t in practice saturates can be determined. This isillustrated in fig. 8, where the resultant N -dependenceof y t is shown for three different W/t = 1 , , W/t = 8 the geometric mean isconverged only for N (cid:38) N (cid:38) W/t = 4; while for
W/t = 1, y t converges for N in excessof 10 . In particular, for what in this context are com-paratively small N (cid:46) y t are sharply increasingwith N for all three disorder strengths. The geometricmean y t (and hence the entire distribution ˜ F ( y )) does30 1 ,
000 2 ,
000 3 ,
000 4 ,
000 5 , N y t W = tW = 4 tW = 8 t ,
000 4 , FIG. 8. For 1BL, geometric means y t = ∆ t /η vs system size N for W/t = 8 (green), 4 (blue) and 1 (red). then converge with increasing N , but the N -values forwhich this is reached in practice are far in excess of any-thing that can be handled in the MBL problem ( N (cid:46) y t , especially for disorderstrengths close to the transition.The full ˜ F ( y ) can in fact be obtained for the one-body d =1 problem. In this case the probability distributionof the local density of states (LDoS) is known exactly inthe thermodynamic limit; specifically the distribution W ( ρ ) of ρ = ρ/ (cid:104) ρ (cid:105) , with ρ the LDoS (at the given en-ergy) and (cid:104) ρ (cid:105) its disorder average. This is given forarbitrary non-zero η (i.e. with Lorentzian broadening ofwidth η for associated δ -functions in the LDoS, as em-ployed in our numerics), and is an inverse Gaussian distri-bution. ∆ I is proportional to the LDoS, ∆ I = cρ (withthe constant c readily shown to be of order t ), so the dis-tribution ˜ F ( y ) of y = ∆ I /η follows as ˜ F ( y ) = ηc W ( ηyc )and is˜ F ( y ) = (cid:114) ξπ y − / exp (cid:104) − ξ ( ηc y − y (cid:105) (53)(specifically with ξ = 4 τ c in weak disorder, and τ themean free time ). Since eq. 53 holds in the thermody-namic limit N → ∞ , the limit of η →
0+ may be takenwith impunity, to give the desired limiting result for thefull distribution ˜ F ( y ). This is obviously˜ F ( y ) = (cid:114) ξπ y − / exp (cid:104) − ξy (cid:105) : η → , (54)which is precisely a L´evy distribution again.Finite- N numerics indeed show clear recovery of thisbehavior. Fig. 9 (upper panel) shows ˜ F ( y ) vs y for awide range of system sizes from N = 20 to 5000 (cf.fig. 5 for the corresponding MBL distributions), withthe L´evy distribution very well captured by N ≈ − − e F ( y ) N = 20 N = 100 N = 400 N = 1200 N = 500010 − − − y e F ( y ) FIG. 9.
Upper panel : For 1BL with
W/t = 4, numericaldistributions ˜ F ( y ) vs y (= ∆ I /η ), for system sizes N rangingfrom N = 20 to 5000 as indicated. The dashed line is a fit ofthe N = 5000 results to a full L´evy distribution eq. 54. Lowerpanel : same data, showing fits (dashed lines) to the inverseGaussian distribution eq. 53 appropriate to non-zero η , whichclearly capture the departures from the leading L´evy tails.The same fit parameters ( ξ and c ) were used throughout, theonly N -dependence arising in η ∝ /N . across essentially the full six-decade y -range shown (savefor the very lowest, which amounts to a negligible frac-tion of the distribution). As seen from the figure, thetails of the finite- N distributions depart from L´evy form( ∝ y − / ) for y ∼ O ( N ) (i.e. ∆ I = ηy ∼ O (1)), in di-rect parallel to the MBL numerics (fig. 5); beyond whichthe L´evy tails are exponentially damped. The latter N -dependent crossover should be captured by the inverseGaussian form eq. 53, since it pertains to finite- η . Thatit clearly does is shown in the lower panel of fig. 9; withthe value of y at which the crossover begins moving toprogressively larger values with increasing N , such thatthe L´evy distribution is recovered in the thermodynamiclimit.With the above in mind, it is natural to ask whetherthe finite- N MBL numerics are similarly captured by theform eq. 53. This indeed appears to be the case, as shownin fig. 10 where the results of fig. 5 are again fit to aninverse Gaussian distribution (and which, though shownhere for
W/t = 20, is found to be representative of theMBL phase). While comparison to eq. 53 is at first sight atwo-parameter fit (in ξ , and c or equivalently ˜ c = c/µ E ),note that if eq. 53 captures the data then the value of˜ c is in practice known, because the mean value of y =∆ I /η = ˜∆ I / ˜ η for the inverse Gaussian eq. 53 is (cid:104) y (cid:105) = c/η = ˜ c/ ˜ η , whence ˜ c = (cid:104) ˜∆ I (cid:105) . As shown in sec. III (fig.1), the arithmetic mean (cid:104) ˜∆ I (cid:105) is well described by theFermi golden rule result eq. 19 in the MBL regime; from410 − − − − − y e F ( y ) N = 10 N = 12 N = 14 N = 161 FIG. 10. Numerical MBL distributions ˜ F ( y ) vs y (= ˜∆ I / ˜ η )for W/t = 20, and system sizes N = 10 −
16 as indicated.Dashed lines show one-parameter fits to the inverse Gaussiandistribution eq. 53, discussed in text. which (together with eqs. 5,9 for Z , µ E )˜ c = (cid:113) π (cid:104) ν (1 − ν ) (cid:0) Vt (cid:1) + (cid:0) Wt (cid:1) (cid:105) − (55)for the band center ˜ ω = 0. In the results shown in fig.10 we have used this N -independent ˜ c (adding that it isfound to be equally satisfactory for W/t (cid:38) N available,fig. 10 shows that the same essential characteristics areseen in the MBL data as for the 1BL case; giving fur-ther support to the result arising from the mean-field ap-proach of secs. IV,V that the limiting ˜ F ( y ) in the MBLphase is a L´evy distribution, at least for the bulk of ˜ F ( y )(the natural caveat again being the lowest y -values, con-vergence of which we believe lies well beyond the acces-sible N -range). B. Delocalized regime
In an extended regime it is as we have emphasized˜∆ I = ∆ I /µ E (with µ E ∝ √ N ) which is perforce non-zero and finite in the thermodynamic limit, rather than∆ I itself. For the mean values (cid:104) ˜∆ I (cid:105) and (cid:104) ∆ I (cid:105) , fig. 1 showsthe former to be well converged with system size for themodest N ’s amenable to calculation; so in consequence (cid:104) ∆ I (cid:105) increases ∝ √ N (fig. 1 inset) for given W/t . Anarithmetic mean is of course merely one reflection of a dis-tribution. The evolution with system size and disorder ofthe geometric mean, ˜∆ t , of ˜∆ I has been considered in fig.4; as found there, it is not yet converged within the avail-able N -range even in the extended regime. This behav-ior is naturally reflected in the full distribution F ( ˜∆ I ),as illustrated in fig. 11 for W/t = 1, well inside the ex-tended phase. F ( ˜∆ I ) appears quite well converged in N for ˜∆ I (cid:38)
1, but some N -dependence clearly remains for 0 0 . . . . e ∆ I F ( e ∆ I ) . . . I N = 10 N = 12 N = 14 N = 161 FIG. 11. Distributions F ( ˜∆ I ) vs ˜∆ I = ∆ I /µ E for W/t = 1and system sizes N = 10 (red), 12 (blue), 14 (green), 16(purple). Inset : corresponding distributions for ∆ I . See textfor discussion. . . . . e ∆ I F ( e ∆ I ) − . . . . FIG. 12. Distributions F ( ˜∆ I ) vs ˜∆ I for W/t = 1 (red),2 (blue) and 4 (green), compared to fits (dashed lines) tothe log-normal form eq. 56. Data for N = 16. Inset : For
W/t = 2, convergence with increasing N of the distribution F ( x ) of x = [ln( ˜∆ I / ˜∆ t )] /σ , to a standard normal distribution(dashed line), shown for N = 12 (blue), 14 (green), 16 (pur-ple). Results for different N are practically indistinguishable. lower values of ˜∆ I (the distribution of ∆ I by contrastshows clear N -dependence for all ∆ I , fig. 11 inset, andas discussed above cannot converge as N → ∞ ). F ( ˜∆ I ) sufficiently deep in the extended phase is in factfound to be of log-normal (LN) form, F ( ˜∆ I ) = 1 √ πσ I exp (cid:16) − [ln( ˜∆ I / ˜∆ t )] σ (cid:17) (56)(with σ the variance of ln ˜∆ I ). This is illustated in fig.12 where comparison of the numerical F ( ˜∆ I ) to eq. 56 isshown for W/t = 1 , , N = 16), the data beingwell captured by the LN distribution for essentially all˜∆ I (including deep in the tails).50 0 .
005 0 .
01 0 .
015 0 . e ∆ I F ( e ∆ I ) − − − − − FIG. 13. Distributions F ( ˜∆ I ) vs ˜∆ I for W/t = 8 (red),10 (blue), 12 (green) and 14 (purple), with N = 16. Inset :same data on log-log scale, with the dotted line showing anemergent ˜∆ − / I component to the tail of F ( ˜∆ I ). Further, given eq. 56 and defining x = [ln( ˜∆ I / ˜∆ t )] /σ ,its probability density F ( x ) should then be of standardnormal form F ( x ) = [2 π ] − / exp( − x / N (and do so rapidly) is shown in the inset tofig. 12 for W/t = 2. Data for
W/t = 1 likewise scalecleanly onto this common form, although by
W/t = 4slight departures from it arise, reflecting the further evo-lution of the distribution with increasing disorder in theextended phase (see below). The occurrence of an F ( ˜∆ I )of LN form for weak disorder seems physically intuitive;and although our simple mean-field approach yields a dif-ferent form (eq. 43), the latter does capture qualitativelythe long-tailed character of F ( ˜∆ I ), with a mode whichdiminishes with increasing disorder.Consider now the evolution of F ( ˜∆ I ) with further in-creasing disorder towards W/t = 12 (the transition oc-curring between
W/t ∼ − probably closerto the upper part of it ). Fig. 13 shows the numeri-cal F ( ˜∆ I ) for W/t = 8 , , ,
14. The distributions re-main unimodal and long-tailed, and with increasing dis-order naturally become increasingly strongly peaked atprogressively lower ˜∆ I . The inset shows the same re-sults on a log-scale, with emphasis as such on the be-havior of the tails in F ( ˜∆ I ). From this, an intermediateregime of L´evy-like power-law behavior ∝ ˜∆ − / I (dot-ted line) is seen to emerge with increasing disorder onapproaching the transition from the extended side (asarises in the mean-field approach, sec. V B); the rangeof which grows in extent with increasing disorder, beforeultimately crossing over to a slower decay.To pursue this further, fig. 14 shows F ( ˜∆ I ) for W/t =12, on a log-scale in the main figure. The natural distri-bution to compare to the numerics is an inverse Gaussian 10 − − − − − e ∆ I F ( e ∆ I ) .
005 0 .
01 0 .
015 0 . FIG. 14. For
W/t = 12, numerical F ( ˜∆ I ) vs ˜∆ I (red line) for N = 16, on a log-log scale. Fits to both an inverse Gaussian(green line), and a L´evy distribution (dotted line) are shown. Inset : same data on a linear scale, showing also the fit to aLN distribution (blue line). Discussion in text. for F ( ˜∆ I ) itself, i.e. F ( ˜∆ I ) = (cid:114) ξ (cid:48) π ˜∆ − / I exp (cid:34) − ξ (cid:48) ( c ˜∆ I − ˜∆ I (cid:35) (57)with ˜ c = (cid:104) ˜∆ I (cid:105) again fixed by eq. 55 (for the same reasonsas given there); and which generates characteristic L´evytails ∝ ˜∆ − / I , before becoming exponentially damped.The inverse Gaussian is compared to numerical results infig. 14 and seen in particular to capture well the tails ofthe distribution, including the departure from the clearlyvisible power-law tail of the corresponding pure L´evy dis-tribution (eq. 57 with ˜ c → ∞ ), also shown (dotted line).While the tail of F ( ˜∆ I ) appears to be described byinverse Gaussian/L´evy, the bulk of the distribution re-sides at lower values of ˜∆ I , as shown in the inset to fig.14 on a linear scale; and which, as seen, is rather wellcaptured by a LN distribution (blue line). The latterseems natural given the clear dominance of the LN formfor lower disorder values (fig. 12); although in the tailregion of F ( ˜∆ I ) the LN distribution decays more rapidlythan inverse Gaussian/L´evy, such that the overall ˜∆ I -dependence of the distribution thus appears to involve acrossover between LN and inverse Gaussian/L´evy behav-iors. VII. CONCLUDING REMARKS
In this paper we have considered many-body localiza-tion in the widely studied model of spinless fermions, on a lattice of N ≡ L d real-space sites. Here the prob-lem has been studied directly from the perspective of the6underlying Fock-space lattice of many-body states; withFS sites coupled by single-fermion hoppings, such thata typical coordination number for a FS site is extensivein N . Such a perspective requires explanation of howan incipient divergence in the FS coordination numberis effectively mitigated, such that an MBL phase evenexists in the thermodynamic limit N → ∞ . Central tothat end, innocuous though it may seem at first sight,has been the rescaling of energy in terms of the standarddeviation of the eigenvalue spectrum µ E ∝ √ N .Exploiting the mapping to a tight-binding model in FS,we have focused specifically on local FS propagators, viatheir associated (Feenberg) self-energies. The imaginarypart ˜∆ I of the rescaled self-energy is of primary impor-tance, being finite in the former case with probabilityunity over an ensemble of disorder realizations, and van-ishingly small for an MBL phase. We have thus focusedon appropriate probability distributions for it; noting inparticular that the geometric mean of the distributioncan act as a suitable order parameter for the transitionto the MBL phase.A self-consistent, probabilistic mean-field approachwas first developed. Despite its simplicity and naturallimitations, this offers physical insight and yields quite arich description of the problem; including recovery of astable MBL phase and an MBL transition in the thermo-dynamic limit, and the notable prediction that the ap-propriate self-energy distribution throughout the MBLphase should be characterized ‘universally’ by a long-tailed L´evy distribution. Informed by the mean-field pic-ture, but free from its assumptions, detailed numericalresults from exact diagonalization in 1 d have also beenpresented. As shown, these provide further detailed in-formation about the underlying probability distributionsin both phases, as well as broad support for the substan-tive predictions arising at mean-field level. ACKNOWLEDGMENTS
Many helpful discussions with John Chalker, H R Kr-ishnamurthy, Sthitadhi Roy and Peter Wolynes are grate-fully acknowledged. We also thank the EPSRC for sup-port, under grant EP/L015722/1 for the TMCS Centrefor Doctoral Training, and grant EP/N01930X/1. Thework is compliant with EPSRC Open Data requirements.
Appendix A
From the definition eq. 14 of the Feenberg self-energy, S I ( ω ) = ω + iη − E I − /G I ( ω ), with the local FS prop-agator G I ( ω ) given in terms of the squared eigenfunc-tion amplitudes | A mI | = |(cid:104) I | Ψ m (cid:105)| (eq. 4) by G I ( ω ) = (cid:80) m | A mI | / ( ω + iη − E m ). From this, the imaginary part of the self-energy ∆ I ( ω ) = − Im S I ( ω ) is∆ I ( ω ) = πD I ( ω )[ G RI ( ω )] + [ πD I ( ω )] − η (A1)with the real/imaginary parts of G I ( ω ) = G RI ( ω ) − iπD I ( ω ) given by πD I ( ω ) = (cid:88) m η | A mI | ( ω − E m ) + η G RI ( ω ) = (cid:88) m | A mI | ( ω − E m ) + η ( ω − E m ) . (A2)Eqs. A1,A2 are used directly in sec. VI to obtain the rel-evant self-energy distributions by exact diagonalization.As mentioned in sec. III, our aim here is to sketch somesimple arguments suggesting that, whether MBL or one-body localization (1BL) is considered, ∆ I ( ω ) is non-zerofor extended states (and proportional to µ E in the many-body case), while by contrast ∆ I ( ω ) ∝ η is vanishinglysmall for localized states. This behavior is also supportedby the numerics of sec. VI. In the following, the site label I refers as usual to the FS site | I (cid:105) ≡ |{ n ( I ) i }(cid:105) in the MBLcase, while for 1BL it implicitly refers to a single real-space site i ( | I (cid:105) ≡ | i (cid:105) = c † i | vac (cid:105) ). Extended states . To make the main points here, weconsider the limit of extended states for weak disorder
W/t (cid:28)
1, on the standard assumption made in thisregime that | A mI | ∼ /N H (for 1BL, N H ≡ N C = N ).With this, eq. A2 for D I ( ω ) gives D I ( ω ) = 1 N H (cid:88) m η/π ( ω − E m ) + η ≡ N H (cid:88) m δ ( ω − E m ) = D ( ω ) , (A3)with D ( ω ) the total density of states/eigenvalue spec-trum. That D I ( ω ) ≡ D ( ω ) is physically clear, since | A mI | ∼ /N H is tantamount to treating all sites I asequivalent in this regime. From eq. A1,∆ I ( ω ) = πD ( ω )[ G R ( ω )] + [ πD ( ω )] (A4)with G R ( ω ) ≡ G RI ( ω ) the Hilbert transform of D ( ω ) (andwhere η ≡
0+ in eq. A1 can clearly be neglected).For 1BL, D ( ω ) is finite throughout the band of single-particle states, whence so too is ∆ I ( ω ). For the MBLcase D ( ω ) is the Gaussian eq. 10, with standard deviation µ E . To be concrete, consider the band center ω = E , forwhich G RI ( ω = E ) = 0 by symmetry. With this, eq. A4gives ∆ I ( ω = E ) = [ πD ( E )] − , whence (from eq. 10)∆ I ( ω ) ∝ µ E . (A5)This result is not confined to the band center; the Hilberttransform of D ( ω ) can be shown to be G R ( ω ) = µ − E × √ F d ( √ ω ) with F d the Dawson function, which fromeq. A4 guarantees eq. A5 for arbitrary ˜ ω = ( ω − E ) /µ E .As above, ∆ I ( ω ) is non-zero for both 1BL and MBL. Inthe former case it is finite, as above. For the many-bodycase (eq. A5), since µ E ∝ √ N , so too is ∆ I ( ω ). Hence˜∆ I (˜ ω ) = ∆ I ( ω ) /µ E remains finite in the thermodynamiclimit N → ∞ . It is thus ˜∆ I ∝ ∆ I / √ N on which onemust focus (this argument being complementary to thatgiven in sec. III). This scaling behavior is also corrobo-rated by the numerical results of fig. 1, where the mean˜∆ I is shown (and is very well converged with system size N by N (cid:39) Localized states . To be specific here, consider ω = E n for some particular localized state | Ψ n (cid:105) with eigenvalue E n . From eq. A2, πD I ( E n ) = | A nI | η + η (cid:88) m ( (cid:54) = n ) | A mI | ( E n − E m ) + η G RI ( E n ) = (cid:88) m ( (cid:54) = n ) | A mI | ( E n − E m ) + η ( E n − E m ) . (A6) There are two potential categories of sites I to be con-sidered: (a) those for which | A nI | vanishes in the ther-modynamic limit, and (b) those for which | A nI | remainsnon-zero in this limit. For the MBL case only the for-mer category arises (as discussed in sec. II); since anMBL state has support on an exponentially large num-ber ∼ N α H ( α <
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The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1993). Further, since the energy E J of all states J under H dif-fers from E I by a finite amount O ( W, V ) (eq. 25), but Z I ∝ N → ∞ , the remaining states of the Wilson chaindecouple completely (the so-called zero-bandwidth limit ofthe Anderson model ). The | (cid:105) -orbital itself thus couplesonly to the ‘impurity’ | I (cid:105) . There are then only two eigen-states in which | I (cid:105) participates, namely the symmetric andantisymmetric combinations of | I (cid:105) and | (cid:105) . Its origins within the present description are easily seen.For low enough λ (where ˜ E I = ( ˜ E I − E ) /µ E may be ne-glected), the local FS propagators and self-energies are es-sentially site-independent, ˜ G I (˜ ω ) ≡ ˜ G (˜ ω ) and ˜ S I (˜ ω ) ≡ ˜ S (˜ ω ); and follow using eqs. 15,21,28 as ˜ S (˜ ω ) = Γ ˜ G (˜ ω ) andhence ˜ G (˜ ω ) = [˜ ω + − Γ ˜ G (˜ ω )] − . The spectrum ˜ D (˜ ω ) followstrivially, π √ Γ ˜ D (˜ ω ) = [1 − (˜ ω / / . Hence ˜∆ t (˜ ω ) hasthe golden rule form of eq. 19, ˜∆ t (˜ ω ) = π Γ ˜ D (˜ ω ), recover-ing the result quoted for the band center ˜ ω = 0. Note that˜ D (˜ ω ) here is semi-elliptic rather than the Gaussian eq. 12,because the problem is treated at second-order RPS level. Reflecting the configurational disorder arising from dis-tributing N e fermions over N sites. For a finite-size system, the band center is taken as thecenter of gravity Tr H = N − s (cid:80) n E n for each disorder re-alization, for the reasons explained in [59]. Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys.Rev. Lett. , 084101 (2013). So η ∼ t/N for W (cid:28) t (with 4 t the full width of thenon-disordered band), while η ∼ W/N for W (cid:29) t . B. Jørgensen,