Rare thermal bubbles at the many-body localization transition from the Fock space point of view
Giuseppe De Tomasi, Ivan M. Khaymovich, Frank Pollmann, Simone Warzel
RRare thermal bubbles at the many-body localization transition from the Fock spacepoint of view
Giuseppe De Tomasi, Ivan M. Khaymovich, Frank Pollmann, and Simone Warzel T.C.M. Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187-Dresden, Germany Department of Physics, Technische Universit¨at M¨unchen, 85747 Garching, Germany Department of Mathematics, Technische Universit¨at M¨unchen, 85747 Garching, Germany
In this work we study the many-body localization (MBL) transition and relate it to the eigen-state structure in the Fock space. Besides the standard entanglement and multifractal probes, weintroduce the radial probability distribution of eigenstate coefficients with respect to the Hammingdistance in the Fock space and relate the cumulants of this distribution to the properties of thequasi-local integrals of motion in the MBL phase. We demonstrate non-self-averaging property ofthe many-body fractal dimension D q and directly relate it to the jump of D q as well as of thelocalization length of the integrals of motion at the MBL transition. We provide an example ofthe continuous many-body transition confirming the above relation via the self-averaging of D q inthe whole range of parameters. Introducing a simple toy-model, which hosts ergodic thermal bub-bles, we give analytical evidences both in standard probes and in terms of newly introduced radialprobability distribution that the MBL transition in the Fock space is consistent with the avalanchemechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBLtransition can been seen as a transition between ergodic states to non-ergodic extended states andput the upper bound for the disorder scaling for the genuine Anderson localization transition withrespect to the non-interacting case. I. INTRODUCTION
Understanding the emergence of ergodicity in closedquantum many-body systems is an active front of re-search [1–6]. Generic interacting systems are expected tothermalize under their own quantum dynamics. Never-theless, thermalization may fail if the system is subjectedto strongly quenched disorder, giving arise to a new phaseof matter dubbed as many-body localized (MBL) [7–13].The MBL phase is best understood in terms of anemergent form of integrability, which is characterized bythe existence of an extensive set of quasi-local conservedquantities, which strongly hinder thermalization in thesystem [14–16]. As a consequence, the system has Pois-son level statistic, area-law entanglement, and the partiallocal structure of the initial state is maintained under theevolution [9, 10, 17]. Instead at weak disorder, the systemis in an ergodic phase, meaning that Eigenstate Thermal-ization Hypothesis [1, 2, 4, 6] (ETH) holds, and thereforelocal observable thermalize. This implies that the systemis fully described in terms of few macroscopic conservedquantities, i.e., energy and/or particles number.A quantum phase transition, referred to as MBL tran-sition [9], is believed to separate an ergodic phase froman MBL one. The MBL transition is a dynamical phasetransition, meaning that it occurs at the level of individ-ual eigenstates even at high energy density. In the lastdecade an enormous effort, both numerically and theoret-ically [7, 17–25], has been made to understand the natureof this transition. Nevertheless, only little is known aboutthe MBL transition. Numerically, the critical exponentsassociated with a putative second-order type of the tran-sition, are in disagreement with generic bounds [17, 26].This incongruence could be due to the fact that the sys- ( a )( b ) ( c )( d ) ↑ ↓ ↑ ↓ ↑ ↑↑ ↑ ↓ ↑ ↓ ↑↑ ↑ ↓ W MBL q , S/S Page
XLL/2 Π (X) 𝑳𝑳 /
𝟐𝟐∼ 𝑳𝑳 pL ∼ L ETH C ritical MBL
FIG. 1. (a) Representation in the Fock space (hypercube) ofthe many-body Hamiltonian (cid:98) H MBL versus the Hamming dis-tance. The blue nodes represent the basis vectors which areconnected by (cid:98) H MBL . (b) Cartoon picture of the radial prob-ability distribution Π( x ) of an eigenstate with respect to thesame Hamming distance x as in the panel (a) in the Fock spacefrom the maximum of the wave function. In the ergodic/ETHphase Π( x ) = L (cid:0) Lx (cid:1) with the maximum at x = L/
2, wheremost of the sites in the Fock space are. In the MBL phase,Π( x ) is skewed on the left with the width ∼ √ L . At the criti-cal point, Π( x ) is much broader and fluctuations are extensivein L . (c) Pictorial representation of the rare thermal bubbles(in red), lengths of which are highly fluctuating at the criticalpoint. The spins in red regions are entangled to each other(shown by many arrows), while the spins away from these re-gions are considered to be frozen (up or down). (d) Sketchof the jump of the entanglement entropy S normalized by itsergodic value S Page and the fractal dimension D q across theMBL transition. tem sizes analyzed using exact diagonalization techniques a r X i v : . [ c ond - m a t . d i s - nn ] N ov are too small to capture the true asymptotic behavior.Very recently, several theoretical works have doubtedthe underline assumptions that the transition is of thesecond order. Phenomenological renormalization groupstudies suggest that the MBL transition could be of aKosterlitz-Thouless (KT) type [27–33]. This possible sce-nario is properly predicted by a possible delocalizationmechanism called avalanche theory, which takes into ac-count non-perturbative effects possibly destabilizing theMBL phase [27, 28]. Strictly speaking, the theory statesthat the presence of thermal bubbles in the system dueto unavoidable entropic arguments is enough to destabi-lize the MBL phase if the localization length ξ loc settledby the disorder strength exceeds a finite critical length.An immediate consequence of this mechanism is that theMBL is characterized not by the divergence of the corre-lation length, as one expects from the ordinary second-order transition, but by a finite jump of the inverse lo-calization length across the transition [28, 30].A complementary interesting perspective is to charac-terize MBL systems in the Fock space, see Fig. 1 (a) fora pictorial representation. This paradigm is based on theoriginal idea of mapping a disordered quantum dot toa localization problem in the Fock space [34] which hasbeen developed further recently [35]. This has been usedto provide evidence of the existence of an MBL transi-tion by Basko, Aleiner, and Altshuler in their seminalwork [7]. Ergodicity is then defined through the fractaldimensions D q , which quantify the spread of a state inthe Fock space [36]. Ergodic states at infinite tempera-tures are believed to behave like random vectors [37, 38],therefore they are spread homogeneously over the entireFock space and D q = 1. Instead, non-ergodic states coveronly a vanishing fraction of the Fock space, 0 ≤ D q < D q = 0,though due to the many-body nature of the problem isnever reached at finite disorder [17, 39–42]. Thus, theMBL transition can be seen as an ergodic to non-ergodictransition in the Fock space, with D q = 1 in the ETHphase and D q < D q for a certainMBL model has been inspected using extensive numer-ical calculations. The MBL transition was found to becharacterized by a jump in the fractal dimensions D q at the critical point. The aforementioned investigationslead to the indication of the existence of an MBL transi-tion. However, a clear connection between the above twoviewpoints is still missing.In this work, we focus on the existence of the abovejump in the fractal dimensions and on its connection tothe avalanche theory, i.e., to the KT-type transition fromanother perspective. Based on the breakdown of self-averaging for D q at the transition and on the recently de-veloped relation of D q to the entanglement entropy [43],we show that the MBL transition is consistent with thejump of D q from D q = 1 to D q < /
2. In addition, wefocus on the radial distribution of a many-body eigen-state in the Fock space around its maximum, relate it to the behavior of local integrals of motion [44], and, thus,confirm the consistency of the KT-scenario [28, 30] forthe MBL transition, see Fig. 1 for an overall picture.This paper is organized as follows. In Sec. II we in-troduce the model and the indicators that we inspectednumerically. In particular, we study the inverse partici-pation ratio of eigenstate coefficients in the Fock space,from which we extract the fractal dimensions and theradial probability of eigenstate coefficients. Section IIIrepresents the numerical results concerning the fractaldimensions and the radial probability distribution. InSec. IV we show our analytical considerations which un-derline the connection between the avalanche theory andthe observed jump in the fractal dimensions.In Sec. V, we provide an example of an non-interactingmodel with many-body filling, which is known to havea delocalization-localization transition and characterizedby a diverging localization length at this transition. Weshow the main difference of this model from the MBLtransition which is believed to have a discontinuity inthe inverse localization length. Finally, we draw our con-clusions and outlooks in Sec. VI.
II. MODEL AND METHODS
We study the random quantum Ising model [44] withthe Hamiltonian (cid:98) H MBL = L (cid:88) i (cid:98) σ xi + L (cid:88) i h i (cid:98) σ zi + V L (cid:88) i J i (cid:98) σ zi (cid:98) σ zi +1 , (1)of a spin chain of the length L with periodic boundaryconditions and the Pauli operators at site i given by σ αi for α ∈ { x, y, z } . h i and J i are independent randomvariables uniformly distributed in [ − W, W ] and [0 . , . W is the disorder and V = 1 is the interac-tion strengths.In [44] under mild assumptions of the absence on en-ergy level attraction the existence of the MBL phasehas been established for sufficiently large, but finite W .Moreover, numerically the critical disorder strength ofthe MBL transition has been identified as W c ≈ . V = 0 the Hamiltonian,Eq. (1), represents a system of uncoupled spins, whichis trivially localized in the sense that all eigenstates areproduct states. In this limiting case, the Hamiltoniancan be expressed, (cid:98) H = (cid:80) i (cid:15) i (cid:98) τ zi , through its integrals ofmotion (cid:98) τ zi = (cid:98) U i (cid:98) σ zi (cid:98) U † i obtained from the original spins bya single-spin rotation (cid:98) U i = (cid:18) cos θ i − sin θ i sin θ i cos θ i (cid:19) , (2)with sin θ i = 1 / (cid:112) h i and the single-spin energies (cid:15) i = (cid:112) h i . This example directly shows that the eigen-states {| τ z (cid:105)} of (cid:98) H MBL with V = 0 are adiabatically con-nected to the σ z -basis product states | σ z (cid:105) = ⊗ i | σ zi (cid:105) with σ zi ∈ {− , } through local rotations (cid:81) i (cid:98) U i | σ z (cid:105) = | τ z (cid:105) .In Ref. 44 it has been shown that for V (cid:54) = 0 and suf-ficiently large W , (cid:98) H MBL can still be diagonalized via asequence of local rotations which adiabatically connectthe eigenstates to the product states in the σ z -basis.Another useful perspective of the model Eq. (1) is toconsider it as an Anderson model on the Fock space. Forthis, one can rewrite the Hamiltonian in the σ z -basis andassociate the first term in Eq. (1) to the hopping and therest to the on-site correlated disorder on a L -dimensionalhypercube, (cid:98) H MBL = (cid:88) σ z ∼ σ (cid:48) z | σ z (cid:105) (cid:104) σ (cid:48) z | + (cid:88) σ z E σ z | σ z (cid:105) (cid:104) σ z | , (3)where | σ z (cid:105) stands for the configuration given by the vec-tor σ z of L values σ zi = { +1 , − } , while σ z ∼ σ (cid:48) z meansthat the corresponding vectors differ by a single spin flip.In this representation the first sum in (cid:98) H MBL can be un-derstood as the Laplace operator on the hypercube as itconnects spins configurations, which differ by one spinflip, and E σ z = (cid:80) i h i σ zi + (cid:80) i J i σ zi σ zi +1 are the diagonalenergies. It is important to note that the 2 L diagonalentries { E σ z } are strongly correlated random variablessince they are constructed only from 2 L random variables { h i } and { J i } . Indeed, even though typical fluctuationsof the entries scales as √ L , their level spacings E σ z − E σ (cid:48) z are O (1) if σ z ∼ σ (cid:48) z .This model should be distinguished from the Quan-tum Random Energy Model (QREM) different from itsclassical counterpart [46–48] for which the diagonal en-tries { E σ z } are replaced by independent identically dis-tributed Gaussian random variables N (0 , W L ) by atransverse field term (cid:80) Li (cid:98) σ xi . The QREM has an An-derson transition at W c ∼ √ L log L (cf. Eq. (10.15) [49]).Ergodic properties of an eigenstate | E (cid:105) of (cid:98) H MBL inEq. (3) can be quantified using the generalized inverseparticipation ratio (IPR q )IPR q = L (cid:88) σ z |(cid:104) σ z | E (cid:105)| q , (4)which quantifies the spread of | E (cid:105) over the Fock space.Through the IPR q the fractal dimensions are defined as D q = log IPR q (1 − q ) L log 2 . (5)Ergodic states at infinite temperature are characterizedby D q = 1 since they extend over the entire Fock space |(cid:104) σ z | E (cid:105)| ∼ / L . In general, 0 < D q < D q = 0 refers to the localized ones.For a model similar to (cid:98) H MBL considered in [40] it hasbeen shown that in the ergodic phase (
W < W c ) mid-spectrum eigenstates show D q = 1. Instead, in the MBLphase D q < D q > W , evendeeply in the MBL phase.The last observation is closely related to the tensorproduct structure of the Fock space and the Hamilto-nian’s local structure. Indeed, in the non-interactinglimit ( V = 0) spins are decoupled, i.e., IPR V =0 q = (cid:81) Li IPR ( i ) q and the one-site IPR ( i ) q is smaller than oneIPR ( i ) q = (cid:88) σ z ∈{↑ , ↓} |(cid:104) σ z | τ zi (cid:105)| q < . (6)As a consequence, IPR V =0 q ∼ − ( q − D q L + O ( √ L ) decaysexponentially with the strictly positive exponent D q = log IPR ( i ) q (1 − q ) log 2 > . (7)This fractal exponent is self-averaging as it is the sumof independent random variables and has fluctuations O (1 / √ L ) shown above in the exponent of the IPR [50].At this point, it is important to appreciate the differ-ence between the MBL phase of (cid:98) H MBL and the localizedphase for the QREM. The first one is characterized bya strictly positive fractal dimension, while in the secondone D q = 0.For a better understanding of the ergodicity propertiesfrom the Fock-space point of view, we define the radialprobability distribution Π( x ) [51] of an eigenstate | E (cid:105) asΠ( x ) = (cid:88) d ( σ z ,σ z )= x |(cid:104) σ z | E (cid:105)| , (8)where the sum runs over the (cid:0) Lx (cid:1) spin states {| σ z (cid:105)} which differ by x flips (i.e., at the Hamming distance d ( σ z , σ z ) = x ) from | σ z (cid:105) , which corresponds to the max-imal eigenstate coefficient max σ z |(cid:104) σ z | E (cid:105)| = |(cid:104) σ z | E (cid:105)| .The overbar indicates the average over disorder and afew mid-spectrum eigenstates.Compared to the IPR q , Π( x ) gives more informationand is a good probe of the eigenstate’s local structure inthe Fock space. In particular, we can study the spreadof Π( x ) by defining the moments X n = (cid:88) x x n Π( x ) , (9)and the mean-square displacement of it∆ X = X − X . (10)In the ergodic phase, the infinite-temperature wavefunction is spread homogeneously on the Fock space andΠ Erg. ( x ) is given by a binomial distribution,Π p ( L, x ) = (cid:18) Lx (cid:19) (1 − p ) L − x p x , (11)with p = 1 / X = L/ X = L/ V = 0), Π( x ) is still given by the binomial probabilitydistribution Eq. (11), however the value of p = sin ( θ i / W as p = 12 − | h i | (cid:112) h i = 12 − √ W + 1 − W . (12)As expected, p (cid:39) / − W/ → / W → V = 0 it happens due to thesystem localization in σ x -basis, and p (cid:39) / (2 W ) → W → ∞ .Note that the binomial probability distribution inEq. (11), can be approximated by a Gaussian distributionwith mean and the variance given by X = pL , ∆ X = p (1 − p ) L . (13)
III. RESULTS
In order to relate the ergodicity properties of the con-sidered system with the local structure of its eigenstatesin the Fock space, we focus on the behavior of the ra-dial probability distribution Π( x ) of mid-spectrum eigen-states of (cid:98) H MBL . However, for sake of completeness westart our analysis by investigating some standard MBLindicators which quantify ergodicity in the real and Fockspace, such as bipartite entanglement entropy and IPR q and compare their properties.The entanglement entropy has been found to be a sur-rounding resource to test and quantify ergodicity in asystem.Figure 2(a) shows the half-chain bipartite entangle-ment entropy S = − Tr[ ρ L/ log ρ L/ ] (blue lines) of thereduced density matrix ρ L/ of a mid-spectrum eigen-states of (cid:98) H MBL . As expected, at weak disorder S showsthe volume law, S ∼ L , and flows towards the Page value S Page = L/ − / S has an area law scaling, S ∼ O (1), and thus S/S
Page ∼ /L tends to zero. Thecrossover between the two behaviors occuring at W c ≈ . D in Eq. (5) is alsoshown in Fig. 2 (a) (orange lines) and its behavior isconsistent with the one of S . In the ergodic phase D ≈ W > W c the fractal dimension converges with L to a value which is strictly smaller than one ( D < S/S
Page and D to unity with the increasing sys-tem size within the ergodic phase and the stabil-ity in the localized phase. The discrete derivatives ∂S = 2 ( S ( L ) − S ( L (cid:48) )) / ( L − L (cid:48) ) log 2 [52] and ∂D = − log (IPR ( L ) / IPR ( L (cid:48) )) / ( L − L (cid:48) ) log 2 in the inset ofFig. 2(a) tend to converge to discontinuous functions of W S / S P a g e , D S/S
Page D ( a ) W ∆ S , ∆ l og I P R ∆ S ∆log IPR ( b ) L = 8101214 W/L − l og I P R ( c ) L = 810121415 V = 0 W/L log L l og I P R / ( d ) W S D W/L r FIG. 2. (a) The averaged bipartite entanglement entropyrescaled by the Page value
S/S
Page (blue) and the averagedfractal dimension D (orange) versus the disorder strength W for (cid:98) H MBL . Its inset shows the discrete derivatives of the en-tanglement entropy ∂S = 2 ( S ( L ) − S ( L (cid:48) )) / ( L − L (cid:48) ) log 2 [52](blue) and ∂D = − log (IPR ( L ) / IPR ( L (cid:48) )) / ( L − L (cid:48) ) log 2(orange) with respect to the system size L . (b) Variances ofthe entanglement entropy S (blue) and of the − log IPR (or-ange) versus W . In both panels the vertical dashed black lineis a guide for eyes indicating the MBL transition. Symbolsshown in panel (b) correspond to different system sizes. Nextpanels show disorder dependence of − log IPR q for (c) q = 2and (d) q = 1 / W ac-cording to the non-interacting limit (15). In both panels, thedashed blue line show the corresponding non-interacting case V = 0. The inset in (d) shows the ratio r -statistics of thelevel spacings versus the rescaled W . W with increasing L , with zero value of ∂S and strictlypositive value of ∂D in the MBL phase. In addition, asclearly seen from Fig. 2(a), both S/S
Page and D deviatefrom their ergodic values at the same disorder amplitude.At the critical point the variance of the entanglement en-tropy ∆ S , which has been shown to be a useful probefor the transition [17, 19], diverges with L (blue linesin Fig. 2(b)). In analogy with S , we also inspect thevariance of − log IP R = LD log 2, which also divergesaround the critical point (orange lines in Fig. 2(b)).The above observations give an indication of the fol-lowing scenario. The upper bound for the unaveraged S ≤ D L log 2 derived in [43] and the deviation of S fromits ergodic Page-value limit S = S Page at the same W as D , is consistent with the jump in D q across the MBLtransition from unity in the ergodic phase to a certainpositive value D q < / D q might lead to the satu- ρ ( D ) ( a ) W = 1 ( b ) W = 2 . L = 81012 1415 D ρ ( D ) ( c ) W = 3 . D ( d ) W = 6 W ∆ D FIG. 3. Probability distribution function ρ of the fractaldimension D for different disorder strength (a) W = 1,(b) W = 2 .
5, (c) W = 3 .
25, (d) W = 6. Different colorsrepresent system sizes L shown in the legend of panel (b).The inset shows the variance of the fractal dimension versus W establishing the non-self averaging property close to theMBL transition (∆ D ∼ O ( L )). The vertical black dashedline in (c) and (d) indicate the value D = 1 / ration of the above bound [43] at the transition pushing S to undergo the volume-to-area law scaling transitionand D q to experience a jump.To obtain more insights in Fig. 3 we analyze the be-havior of the fractal dimension fluctuations via the prob-ability distribution of D for several relevant values of W . Deeply in the ergodic phase D tends to unityand its fluctuations are exponentially suppressed with L ,as dictated by ETH at infinite temperature (Fig. 3(a-b)). Instead, in the localized phase (Fig. 3(d)), weexpect D < / D ∼ /L as shown in the inset of Fig. 3(a). The self-averaging of D q in the MBL phase can be understood inits non-interacting limit ( V = 0),∆ D q = (cid:80) i ∆(log IP R ( i ) q ) L (1 − q ) log ∼ L . (14)On the contrary, in the critical region ( W ≈ W c ), wefind that D q is not self-averaging since its fluctuation donot decay with L (inset of Fig. 3) and the probabilitydistribution is stuck and not shrinking with increasing L (Fig. 3(c)).At critical points of single-particle problems, self-averaging is usually absent if D q demonstrates ajump [36]. Its absence is far from being trivial in many-body problems and it provides another case for the jump of D q at the critical point. Summarizing, at the MBLtransition D q might be characterized by a jump and it isnot self-averaging. This non-self-averaging and the rela-tion between D q and S [43] might drive the simultaneousjump-like transitions in S and D q .It is important to point out once again that D q isstrictly positive due to the many-body nature of theproblem. As a consequence, the MBL transition cannotbeen considered as an Anderson-localization transitionin the Fock space with D q = 0 in the localized phase,but rather as a transition between an ergodic ( D q = 1)and a non-ergodic extended (0 < D q <
1) phase. In or-der to reach a genuine localization transition in the Fockspace one needs to rescale the disorder strength with L .Naively, as a first approximation, one might replace theFock space on-site energies with independent distributedrandom variables with typical fluctuation ∼ √ L as isdone for the QREM [53–57]. If this would be the case,the Anderson transition would occur at W AT ∼ √ L log L also for the quantum Ising model Eq. (1). Neverthe-less, the Fock space on-site energies are strongly corre-lated and they cannot be approximated as independentrandom variables. Due to the presence of these correla-tions (similar to [58]) we expect that stronger disorder isneeded to localize many-body states in the Fock space.In order, to understand the correct scaling of W AT with L , we rely on the exactly solvable non-interactingcase (c V = 0) providing the lower bound for D q andthen check numerically if the same scaling works as wellat strong disorder in the interacting case. This compari-son is motivated by the belief [44] that interacting eigen-states in the MBL phase are adiabatically connected tothe ones of a non-interacting problem. Consequently, weexpect to rescale W with L in the same way as in thenon-interacting problem in order to have a genuine local-ization in the Fock space. Straightforward calculationsof the single-spin IPR ( i ) q = sin θ i + cos θ i in the non-interacting model show that at large disorder WD q ∼ W − q q < / , log WW q = 1 / ,W − q > / . (15)Thus rescaling W ∼ L , we have IPR q ∼ O (1) for q > / − log IPR for V = 1 which is collapsedafter the rescaling W/L in agreement with the predictionof the non-interacting calculation.The last observation should be compared to the scalingof W for the QREM, where W QREM ∼ √ L log L (cid:28) L and the random many-body energies are uncorrelated.However, due to the limitation of system sizes achievablewith exact diagonalization techniques, it is important tonotice that a reasonable collapse of the curves in Fig 2(c)can also be obtained by rescaling W/ ( √ L log L ). Thus,in order to draw a distinction between the two scalings,we consider another moment q = 1 / q . Thecritical value for the QREM is independent of q , whilefor the MBL model we expect a different scaling with L for q ≤ /
2, as shown in Eq. (15). Figure 2(d) clearlydemonstrates the collapse of log IPR / for several L , byrescaling W with L log L in complete agreement with theprediction of Eq. (15).Recently, some works [59, 60] claim that the criticalpoint for the MBL transition is L -dependent and shiftsas W c ∼ L . These claims are in conflict with our resultsfor which the system is already Anderson-localized in theFock space ( D q = 0) provided W ∼ L . Localization inthe Fock space is a much stronger breaking of ergodicitythan the one defined using local observables (ETH). Sincewe expect to have D q = 1 in a putative ergodic phase atinfinite temperature, it implies that the critical point ofthe MBL transition, if it scales with system size, wouldhave to scale slower than L ( W c /L → W c with L (if any). It is natural to expect that the stronger the in-teraction is, the stronger is the disorder needed to breakthe ergodicity and reach MBL. From this perspective, theuncorrelated on-site disorder in QREM can be expressedin terms of strings of (cid:98) σ z operators L (cid:88) σ z E σ z | σ z (cid:105) (cid:104) σ z | = L (cid:88) k (cid:88) i ,..,i k J ( k ) i ,...,i k (cid:98) σ zi · · · (cid:98) σ zi k , (16)with J ( k ) i ,...,i k = L (cid:80) σ z E σ z σ zi · · · σ zi k . As compared withHamiltonian (cid:98) H MBL , the QREM contains stronger inter-actions, and, for the system Eq. (1), we expect W c ≤ W QREM ∼ √ L log L < W AT ∼ L . (17)This imposes an even stronger upper bound on the pos-sible scaling of the MBL transition with L , which is notconsistent with [59, 60]. A. Radial probability distribution Π( x ) In this section, we show that the radial probabilitydistribution Π( x ) in Eq. (8) of mid-spectrum eigenstatecoefficients of (cid:98) H MBL in the Fock space gives more detailedinformation on the wave function’s local structure in theFock space compared to IPR q . In addition, it can berelated to the local integrals of motion of (cid:98) H MBL [14–16, 44].We recall that at strong disorder, deep in the MBLphase, the eigenstates {| E (cid:105)} of (cid:98) H MBL are believed tobe adiabatically connected to the non-interacting ones {| σ z (cid:105)} through a sequence of quasi-local unitary opera-tors, | E (cid:105) = (cid:98) U | σ z (cid:105) , (18)defining the integrals of motions (cid:98) τ zi = (cid:98) U (cid:98) σ zi (cid:98) U † for which[ (cid:98) H, (cid:98) τ zi ] = 0. x Π ( x ) ( a ) W = 1 Π Erg ( x ) W › X fi / L ( b ) Erg L = 810121416 V = 0 W ∆ X / L ( c ) W -1 H g W Γ b FIG. 4. (a) Radial probability distribution Π( x ) of the eigen-state coefficients in the Fock space, Eq. (8), for a fixed systemsize L = 16 and several disorder strengths W , ranging fromthe ergodic to the MBL phase (see legend). The blue dashedline show the ergodic prediction for Π Erg ( x ) = L (cid:0) LL/ (cid:1) . (in-set) shows the Hellinger distance H g , Eq. (25), between Π( x )and the ergodic one Π Erg ( x ). (b) Mean X/L , Eq. (9), and(c) variance ∆ X /L , Eq. (10), of Π( x ) as functions of W forseveral L . Symbol and color code is shown in legend. Theblue and green dashed lines correspond to the ergodic andthe non-interacting ( V = 0) cases, respectively. The insetshows the deviation of Π( x ) from a binomial form (11) viathe ratio Γ b = L ∆ X /X ( L − X ). Now, we use the above assumption to find the relationbetween the spread of the local integrals of motion { (cid:98) τ zi } and the moments of the probability distribution Π( x ).The Hamming distance between two Fock-states | σ z (cid:105) and | σ z (cid:105) is given by d ( σ z , σ z ) = (cid:88) i ( σ zi − σ z ,i ) , (19)where σ zi = (cid:104) σ z | (cid:98) σ zi | σ z (cid:105) . The first moment of Π( x, E ),Eq. (9), for a certain eigenstate at energy E is given by, X ( E ) = (cid:88) x x Π( x, E ) = (cid:88) σ d ( σ z , σ z ) (cid:12)(cid:12)(cid:12)(cid:68) σ z (cid:12)(cid:12)(cid:12) (cid:98) U (cid:12)(cid:12)(cid:12) σ z (cid:69)(cid:12)(cid:12)(cid:12) , (20)where we use | E (cid:105) = U | σ z (cid:105) . Thus, X ( E ) = L − (cid:88) i (cid:88) σ z σ zi σ z ,i (cid:68) σ z (cid:12)(cid:12)(cid:12) (cid:98) U † (cid:12)(cid:12)(cid:12) σ z (cid:69) (cid:68) σ z (cid:12)(cid:12)(cid:12) (cid:98) U (cid:12)(cid:12)(cid:12) σ z (cid:69) = L − (cid:88) i (cid:104) E | (cid:98) σ zi (cid:98) τ zi | E (cid:105) . (21)Averaging over disorder and energies E , we obtain X = L − (cid:88) i (cid:104) E | (cid:98) σ zi (cid:98) τ zi | E (cid:105) (22)Similar calculations show that the variance in Eq. (10) ofΠ( x, E ), ∆ X ( E ) = X ( E ) − X ( E ) , is given by∆ X ( E ) = 14 (cid:88) i,j (cid:10) E (cid:12)(cid:12)(cid:98) σ zi (cid:98) σ zj (cid:98) τ zi (cid:98) τ zj (cid:12)(cid:12) E (cid:11) −− (cid:32)(cid:88) i (cid:104) E | (cid:98) σ zi (cid:98) τ zi | E (cid:105) (cid:33) . (23)Thus, both first cumulants, X and ∆ X , provide themeasures of the distance between the local integrals ofmotion { (cid:98) τ zi } and the undressed operators { (cid:98) σ zi } . In par-ticular, in the MBL phase we expect a perturbativeexpansion τ zi = a αi (cid:98) σ αi / √ ξ loc + b αβi i (cid:98) σ αi (cid:98) σ βi + · · · [61],where a αi ∼ e −| i − i | /ξ loc , b αβi i ∼ e − ( | i − i | + | i − i | ) /ξ loc andLatin letters run over site indices and Greek ones are in { x, y, z } . The coefficient 1 / √ ξ loc , with a αi ∼ O (1), pro-vides the normalization of the operator ( τ zi ) = 1. Thus, X gives the direct estimate for the localization length X ∼ L (cid:32) − (cid:115) ξ min ξ loc (cid:33) , (24)where ξ min stands for some typical value of ( a αi ) .Having elucidated the relation between the integralsof motion and the radial probability distribution, we nowpresent the numerical results for Π( x ), X and ∆ X . Fig-ure 4(a) shows Π( x ) for fixed system size L = 16 and sev-eral disorder strengths. As expected, at weak disorder,Π( x ) is centered in the middle of the chain, X = L/ X = L/ x ) is close to the one Π Erg ( x )of an ergodic system, Eq. (11) with p = . To quantifybetter the deviations of Π( x ) from Π Erg ( x ), we considerthe Hellinger distance H g = 1 √ (cid:118)(cid:117)(cid:117)(cid:116)(cid:88) x (cid:18)(cid:112) Π( x ) − (cid:113) Π Erg ( x ) (cid:19) , (25)which quantifies the distance between two probabilitydistributions. The inset in Fig. 4(a) shows H g as a func-tion of W . In agreement with the IPR q -analysis, at weakdisorder H q tends to zero exponentially fast with L , sincethe system is ergodic in the Hilbert space. At strong dis-order in the MBL phase, Π( x ) has a non-ergodic shapeand it is skewed to the left with respect to its maximum,meaning that X < L/
2, Fig. 4(a), and H g flows slowlywith L to its maximal value H g = 1.Both mean X/L and variance ∆ X /L normalized by L in Fig. 4(b-c), demonstrate similar behavior to thefractal dimension, D q , and the normalized entanglement entropy, S/L and its fluctuations shown in Fig. 2(a-b). The quantity
X/L decreases monotonically with W from its ergodic value (blue dashed line) toward the non-interacting one (green dashed curve). At W (cid:46) W c , X/L flows with L towards the ergodic value X/L = 1 /
2, con-sistent with ξ − = 0 in Eq. (24). It shows the saturationwith the system size in the MBL phase at the strictly pos-itive value ( ξ loc > ξ min ). This behavior is consistent withthe finite jump in ξ − at the MBL transition. The devi-ations of X/L and ∆ X /L in the MBL phase from theirnon-interacting limits and the saturation of these devi-ations with the system size are possibly related to thepartial melting of the ergodic bubbles in the avalanchescenario [28].Close to the transition ∆ X /L exhibits a peak whichdiverges with the system size, ∆ X ∼ L α , with α > x ) goingbeyond the binomial approximation, Eq. (11). The insetto Fig. 4(c) emphasizes this by showing that the ratioΓ b = L ∆ X /X ( L − X ) is strictly larger than its binomialunit value and demonstrates the same divergence with L as the variance ∆ X /L .To sum up in this section we have studied the radialprobability distribution Π( x ), which is directly related tothe local integrals of motions. In particular, the mean X of Π( x ) can be used to define a localization length ξ loc ,Eq. (24). We have shown that at weak disorder X → L/ ξ loc → ∞ , while in the localizedphase 0 < X/L < / ξ loc > ξ min . At thetransition the fluctuations ∆ X diverge which gives anevidence of a finite jump in X/L and therefore in ξ − inthe thermodynamic limit L → ∞ . The above finite jumpin ξ − is consistent with the avalanche theory of many-body delocalization [28] and therefore with the KT-typescaling of the transition [29, 30].In the next section, we consider a simple toy modelof dilute ergodic “bubbles”, which explains the observedphenomenon of the absence of D q self-averaging, andhelps us to bridge the gap between Fock space structureof mid-spectrum eigenstates, the avalanche theory, andthe recent studies of phenomenological renormalizationgroups of MBL transition. IV. DILUTE ERGODIC BUBBLESAPPROXIMATION
In this section we discuss random block Hamiltonianconsisting of non-interacting pieces which is able to repro-duce the numerical results described in the previous sec-tion. Ref. 28 proposes that non-perturbative effects, suchas rare thermal inclusions, which are unavoidable dueto entropic arguments, can destabilize the MBL phaseprovided their density exceeds a certain critical value.As a consequence, an abrupt finite jump of the local-ization length is expected at the transition. Moreover,RG-studies [29, 30] have shown that this avalanche the-ory should lead to a KT-like scaling behavior and theprobability distribution P ( (cid:96) ) of the size (cid:96) of the largestthermal bubble has a power-law fat tail leading to a di-verging variance.With the aim to keep the toy model as simple as pos-sible and avoid introducing further “fitting” parameters,we consider bimodal approximation of ρL active ergodicspins [62] and (1 − ρ ) L non-interacting (frozen) spins. AsΠ( x ) is not sensitive to the phases of the eigenstate’s co-efficients, and both ergodic and non-interacting limits ofit are given by the binomial distribution in Eq. (11), weapproximate the contributions of ergodic and frozen spinsto the many-body wave function as the non-interactingones (Eq. (2)), with sin θ i = for ρL active spins andsin θ i = 1 / (cid:112) h i for the remaining (1 − ρ ) L frozenones, as they are subject to strong fields h i . Due to self-averaging of the non-interacting contribution (14) we de-scribe it by the mean value of p , Eq. (12). In this model, ρ plays the rote of the density of ergodic bubbles.Within the above approximation, one can easily findΠ( x ) as follows. At a spin-flip distance x from the eigen-state maximum we take k spin flips of ergodic type and r − k of frozen type. The number of such paths from themaximal configuration point in the Fock space is givenby N k,x = (cid:18) ρLk (cid:19) (cid:18) (1 − ρ ) Lx − k (cid:19) , (26)where the first combinatorial factor counts the numberof possible selection of k indistinguishable spin-flips ofthe ergodic type out of ρL available ones, the secondone counts the number of possible r − k indistinguishablespin-flips of the frozen type out of (1 − ρ ) L available ones.The corresponding wave function intensity averaged over h i will be | ψ k,x | = (cid:18) (cid:19) ρL p x − k (1 − p ) (1 − ρ ) L − x + k . (27)The total number of selections for any k is given by N x = x (cid:88) k =0 N k,x = (cid:18) Lx (cid:19) . (28)As a result, one can obtain the probability distributionΠ( x, ρ ) at the fixed ergodic spin density ρ Π( x, ρ ) = x (cid:88) k =0 N k,x | ψ k,x | = x (cid:88) k =0 Π / ( ρL, k ) Π p ((1 − ρ ) L, x − k ) (29)which is simply given by the convolution of two binomialdistributionsΠ p (cid:48) ( L (cid:48) , x (cid:48) ) = (cid:18) L (cid:48) x (cid:48) (cid:19) p (cid:48) x (cid:48) (1 − p (cid:48) ) L (cid:48) − x (cid:48) , (30) one describing the ergodic bubble ( p (cid:48) = 1 /
2) and theother the remaining frozen spins with p (cid:48) = p < / q isthe product of the IPR q s for the ρL ergodic spins andthe (1 − ρ ) L frozen ones:IPR q = (cid:88) x,k N k,x | ψ k,x | q == (cid:18) (cid:19) ( q − ρL [ p q + (1 − p ) q ] (1 − ρ ) L , (31)and thus the fractal dimensions, Eq. (5), take the form D q ( ρ ) = ρ + (1 − ρ ) D q , (32)with D q the fractal dimension of the non-interactingfrozen spins, which is a continuous function of W withEqs. (7), (15). In this approximation any jump in thefractal dimension D q ( ρ ) is directly related to the jumpin the bubble density ρ .By using the Gaussian approximation Eq. (13) for thebinomial distribution Eq. (11), we can estimate the firstand second moment of Π( x, ρ ) in Eq. (29) X ρ = L − νεL, (33) X ρ = (cid:20) L − νεL (cid:21) + L (cid:20) − νε (cid:21) (34)and ∆ X ρ = X ρ − X ρ = L (cid:20) − νε (cid:21) , (35)where ε = 1 / − p and ν = 1 − ρ ≤
1. These resultshold for fixed density ρ . For ρ = 0 and p given by (12),they coincide with the non-interacting results shown inFig. 4 (b-c) as green dashed lines, which show plots ofEqs. (33) and Eq. (35) for these parameters.We emphasize that the ratio Γ b ( ρ ) = L ∆ X ρ /X ρ ( L − X ρ ) is strictly smaller then its binomial value Γ b in case0 < ν < ε (cid:54) = 0, and X ρ L (cid:0) L − X ρ (cid:1) = L (cid:20) − ν ε (cid:21) , (36)which contradicts the numerics in Fig. 4. This observa-tion underlines the importance of the fluctuations of ρ and the corresponding disorder average. Moreover, eventhe scaling of the maximum of ∆ X ∼ L α , α >
1, fromthe exact numerics with L contradicts (35) with fixed ρ .In order to recover the observed behavior of Γ b > P ( ρ ) of the density of ergodic spins and per-form the average over it. After averaging Eqs. (32), (33),and (34) and abbreviating (cid:90) ρP ( ρ ) dρ = ¯ ρ ≡ − ¯ ν, (cid:90) ρ P ( ρ ) dρ − (¯ ρ ) = σ ρ , (37)one straightforwardly obtains¯ D q = 1 − ¯ ν (1 − D q ) , (38) X = L − ¯ νεL (39)∆ X = σ ρ ε L + L (cid:20) − ¯ νε (cid:21) . (40)The properties of the probability distribution P ( ρ ) ofthe density of ergodic spins can be extracted from theworks [27–30]. According to these works the distribu-tion P ( (cid:96) i ) of the length (cid:96) i of a single ergodic bubbleis (stretch)-exponential, P ( (cid:96) i ) ∼ e − c(cid:96) di , or power-law, P ( (cid:96) i ) ∼ (cid:96) i − α , α > P ( (cid:96) i ) is sharply con-centrated on the length scale L of the system size andtends to a delta-function for L → ∞ , meaning the en-tire system is ergodic. At the transition P ( (cid:96) i ) is apower-law distributed ∼ (cid:96) − α c i with the critical exponent2 ≤ α c ≤ (cid:96) ∼ L anddiverging variance as (cid:96) ∼ L − α c .In our consideration we focus on the power-law prob-ability distribution and determine the power α indepen-dently. Due to entropic reasons, the mean density ρ ofthe ergodic bubbles has to be finite even in the MBLphase. This density can be approximated by the sum ofindividual bubbles of lengths (cid:96) i normalized by the systemsize ρ ≈ K (cid:88) i =1 (cid:96) i L , (41)where K is a cut-off ensuring finite total length of order L of the chain. In the simplest approximation of inde-pendent identically distributed random (cid:96) i one can applythe central limit theorem to ρ .In the MBL phase where the mean and the variance ofthe length of a single ergodic bubble are finite, (cid:96) ∼ L , (cid:96) ∼ L , i.e., α >
3, and the number of bubbles scalesas K ∼ L due to the boundness of ρ = K(cid:96)/L ∼ L , oneobtains the Gaussian distribution of ρ with the variancedecaying as σ ρ ( W > W c ) = ¯ ρ − ¯ ρ ∼ L , (42)consistent with self-averaging. This also immediatelyconfirms the self-averaging of D q via Eq. (32). In aergodic phase, the variance σ ρ ( W < W c ) is exponen-tially small in L as ρ →
1. At the MBL transitionself-averaging of ρ breaks down in this model only if (cid:96) → L − α c diverges in the thermodynamic limit, i.e. α c <
3. Due to the finite mean density ρ = K (cid:88) i =1 (cid:96) i L = KL max (cid:0) L , L − α c (cid:1) ∼ L , (43)the number of ergodic bubbles scales as K ∼ min (cid:0) L, L α c − (cid:1) , which puts the lower bound on α c >
1. The corresponding scaling of the variance takes the form σ ρ ( W (cid:39) W c ) ∼ min (cid:0) L , L − α c (cid:1) , < α c < . (44)The condition of ∆ X scaling faster than L is in agree-ment with the observations in Fig. 4(c), and requires α c <
3, which ultimately leads to the decay of the vari-ance with L , but slower than L − . As a consequence,both in the MBL phase and at the transition the mostsignificant contribution to ∆ X in (40) is given by thelarge fluctuations in ρ , cf. Eqs. (42), (44).The lack of self-averaging of D q in Eq. (32) implies themore restrictive condition α c ≤
2. This is consistent withthe finite mean (cid:96) ∼ L of a single bubble, α c ≥
2, onlyat the value α c = 2.To summarize, in the ergodic phase ρ = 1 − ν goes tounity exponentially ∼ e − ηL and its variance has to de-cay also at least exponentially and therefore ∆ X ( W (cid:28) W c ) (cid:39) L/
4, as expected in the ergodic case. In the MBLphase ρ is finite and the variance decays as 1 /L (cf. (42)),which yields a linear behavior of the mean X ∼ L (cf. Eq. (39)), and the variance ∆ X ∼ L (cf. Eq. (40)).At the transition, W = W c , according to Eq. (44), thevariance of ρ is large compared to L − and, thus, it pro-vides the main contribution to ∆ X ∼ L − α c (cid:29) L , with2 ≤ α c ≤
3, i.e.∆ X /L ∼ / W < W c ,L − α c W = W c ,< / W > W c . (45)Finally, the constraint α c = 2 implies also that D q is notself-averaging at the transition.Thus, we have shown that our numerical results can beexplained by considering a simple toy model of dilute er-godic bubbles embedded into a sea of frozen clusters. Inparticular, the divergence in ∆ X /L requires to describethe probability distribution of the length of a typical er-godic bubble by a fat-tailed power-law distribution whichhas diverge fluctuations at the transition,∆ D q ∼ σ ρ ∼ e − ηL W < W c ,L − α c W = W c ,L − W > W c . (46) V. CONTINUITY OF THE TRANSITION FORNON-INTERACTING SYSTEMS
In the previous sections, we have considered numeri-cally the MBL transition from a Fock-space perspective.We have shown that it is could be characterized by a fi-nite jump in the fractal dimensions D q , which are notself-averaging at the critical point. This scenario is con-sistent with the avalanche theory which predict a KT-type scaling at the transition and therefore a finite jumpon the inverse localization length ξ − . For a contrast, wenow investigate the case in which the system undergoes0 W D ξ − loc = log W/ ( a ) W L ∆ D ( b ) L = 128192256384512 log ξ loc D FIG. 5. (a) Fractal exponent D and (b) its variance ∆ D forthe non-interacting Aubry-Andr´e model with the many-bodyfermionic half-filling for several L versus the amplitude of thequasi-periodic potential W . The vertical black dashed linepoints to the critical point W c = 2. The y -axis of panel (b) hasbeen rescaled to show that D is self-averaging (∆ D ∼ /L ).The inset shows D for W > ξ − = log W/ a delocalization-localization transition with continuous ξ − , corresponding to the divergence of the localizationlength at the transition, lim W → W c ξ loc → ∞ . Concretely,we consider the non-interacting Aubrey-Andr´e model (cid:98) H AA = − (cid:88) i (cid:98) c † i +1 (cid:98) c i + h.c. + W (cid:88) i h i (cid:98) c † i (cid:98) c i , (47)where (cid:98) c † i ( (cid:98) c i ) is the fermionic creation (annihilation) op-erator at site i , h i = cos (2 πβi + δ ) is a quasi-periodicpotential with β = ( √ /
2, and δ is a random phaseuniformly distributed in the interval [0 , π ). This single-particle model is known to have an Anderson transi-tion at W c = 2 between extended ( W <
2) and local-ized (
W >
2) phases [63]. The single-particle localiza-tion length at transition diverges as ξ loc ∼ / log W/ (cid:98) H AA with the many-bodyfermionic filling and focus on multifractal structure ofthe Slater-determinant many-body wave functions in theFock space. For this reason, we take the model on L sites in the chain at half-filling N = L/
2, where N is thenumber of particles, maximizing the Hilbert space dimen-sion (cid:0) LN (cid:1) ∼ L / √ L . Fixing the basis of bare fermions | n (cid:105) = (cid:81) n i ( (cid:98) c † i ) n i | (cid:105) as the computational basis, with n i ∈ { , } , we rewrite the model (cid:98) H AA in a similar formas Eq. (3) and investigate the ergodic properties of itseigenstates in the Fock space.Figure 5(a) shows the fractal dimension D for theeigenstates of (cid:98) H AA constructed as a Slater determinantof taken at random N single-particle eigenstates. In theextended phase, W <
2, the fractal dimension tends tounity, meaning that the typical eigenstate covers homo-geneously the available Hilbert space. Instead, in thesingle-particle localized phase,
W > D converges toa strictly positive value smaller than one. In fact, we ex-pect that IP R ∼ ( ξ loc ) L similarly to a non-interacting spin chain, Eq. (7), and as a consequence D ∼ log ξ loc close to the transition. Inset of Fig. 5(a) shows D as function log ξ loc giving an evidence of our predictionand therefore the many-body D does not experience ajump across the transition. Another important conse-quence is that D is self-averaging as shown in Fig. 5(b),∆ D ∼ /L , everywhere including the transition. Thisself-averaging property should be compared to the oneobserved in the MBL model, for which the fractal dimen-sions are not self-averaging close to the MBL transition. VI. CONCLUSION
In this work we have studied the MBL transition ofa chain of interacting spins from a Fock-space point ofview. In addition to the standard diagnostic tools forergodicity, such as fractal dimensions and entanglemententropy, we consider the radial probability distributionof eigenstate coefficients with respect to the Hammingdistance in the Fock space from the wave function max-imum. We show that this radial probability distributiongives important insights about the integrals of motion ofthe problem and allows to extract the localization lengthfrom the cumulants of the distribution.Numerically we have found that both the fractal di-mensions and the radial probability distribution havestrong fluctuations at the critical point. The fractal di-mensions at the transition are not self-averaging and theprobability distribution is extremely broad. This diver-gence provides a rather strong evidence of the existenceof a possible jump of the fractal dimensions as well as ofthe localization length across the MBL transition.Inspired by recent studies of renormalization group andavalanche MBL theory, we explain our findings by in-troducing a simple spin toy model. This model hostsa finite density of ergodic/thermal bubbles as well asfrozen/localized spins. The MBL transition occurs bytuning the density of the ergodic bubbles. At the transi-tion the probability distribution of the bubble density isfat-tailed in agreement with the RG studies. Using thissimple model, we are able to explain our numerical find-ings, and thus to bridge the gap between recent studiesof the nature of the MBL transition in the “real space”and in the Fock space perspectives.As a result, we show that the MBL transition can beseen as a transition between ergodic states to non-ergodicextended states and we put an upper bound on the disor-der scaling for a genuine Anderson localized regime withrespect to the non-interacting case.Finally, we provide an example of the many-body (non-interacting) model with a continuous localization transi-tion showing the self-averaging of fractal dimensions inthe whole range of parameters. This model confirms theconjectured relation between the non-self-averaging prop-erty of the fractal dimensions and their finite jump at thelocalization transition.1
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