Eigenstates hybridize on all length scales at the many-body localization transition
EEigenstates hybridize on all length scales at the many-body localization transition
Benjamin Villalonga ∗ and Bryan K. Clark † Institute for Condensed Matter Theory and IQUIST and Department ofPhysics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (Dated: May 29, 2020)An interacting quantum system can transition from an ergodic to a many-body localized (MBL)phase under the presence of sufficiently large disorder. Both phases are radically different in theirdynamical properties, which are characterized by highly excited eigenstates of the Hamiltonian.Each eigenstate can be characterized by the set of quantum numbers over the set of (local, in theMBL phase) integrals of motion of the system. In this work we study the evolution of the eigenstatesof the disordered Heisenberg model as the disorder strength, W , is varied adiabatically. We focuson the probability that two ‘colliding’ eigenstates hybridize as a function of both the range R atwhich they differ as well as the strength of their hybridization. We find, in the MBL phase, that theprobability of a colliding eigenstate hybridizing strongly at range R decays as P r ( R ) ∝ exp[ − R/η ],with a length scale η ( W ) = 1 / ( B log( W/W c )) which diverges at the critical disorder strength W c .This leads to range-invariance at the transition, suggesting the formation of resonating cat statesat all ranges. This range invariance does not survive to the ergodic phase, where hybridization isexponentially more likely at large range, a fact that can be understood with simple combinatorialarguments. In fact, compensating for these combinatorial effects allows us to define an additionalcorrelation length ξ in the MBL phase which is in excellent agreement with previous works andwhich takes the critical value 1 / log(2) at the transition, found in previous works to destabilize theMBL phase. Finally, we show that deep in the MBL phase hybridization is dominated by two-levelcollisions of eigenstates close in energy. PACS numbers: 75.10.Pq,03.65.Ud,71.30.+h
Introduction.—
Certain disordered interactingsystems can transition to the many-body localized(MBL) phase, breaking ergodicity and defying ther-malization in the process [1–7]. A fully MBL Hamil-tonian, like the sufficiently disordered one-dimensionalHeisenberg model (Eq. (2)), can be diagonalized by aunitary operator U which can be expressed as a low-depth quantum circuit [8–11]. This results in localizedeigenstates with area law entanglement [12–14], i.e. ,they can be disentangled to product states througha small number of local transformations. By con-tinually decreasing the strength of the disorder, thesystem’s eigenstates evolve transitioning into ergodiceigenstates at a sharp value of the disorder strength;these eigenstates are delocalized and have volume lawentanglement. Furthermore, while MBL eigenstatesdo not experience level repulsion and follow Poissonstatistics, ergodic eigenstates follow GOE [6, 15, 16].For recent reviews on MBL see Refs. [5, 17–22].For a spin- system whose Hamiltonian H is diag-onalized by the unitary matrix U , the two-level Her-mitian operators τ zi ≡ U σ zi U † (1)commute with H and with each other. In MBL, theseoperators (often called (cid:96) -bits) have local support andcan be interpreted as locally dressed spins. (cid:96) -bits havebeen a key quantity in understanding the phenomenol-ogy of the MBL phase [8–11], and considerable effort ∗ [email protected] † [email protected] has been devoted to numerically access (cid:96) -bits [11, 23–29]. Using (cid:96) -bits we can identify each eigenstate | n (cid:105) of H via the L pseudo-spins corresponding to theeigenvalues of the L (cid:96) -bits { τ iz } . Alternatively, theseeigenvalues correspond to the spin configuration of theproduct state generated by U † | n (cid:105) .We focus on the one-dimensional spin- nearest-neighbor antiferromagnetic Heisenberg chain withrandom on-site magnetic fields: H ( W ) = 14 L − (cid:88) i =0 (cid:126)σ i · (cid:126)σ i +1 − W L − (cid:88) i =0 h i σ zi . (2)The on-site magnetic fields { h i } are sampled uni-formly at random from [ − ,
1] and W is the disor-der strength. The model of Eq. (2) has been studiedextensively in the context of MBL [6, 7, 12, 13, 15, 30–46]. It presents an ergodic-MBL transition at infinitetemperature (middle of the energy spectrum) at a crit-ical W c ≈ . W that varies with energy, forming a mo-bility edge [6, 47]. This Hamiltonian conserves totalmagnetization (cid:80) i σ zi ; throughout this work we focuson the zero magnetization sector.While many aspects of the MBL phase are wellunderstood, much less is known about the transi-tion, where locality must break down. Renormaliza-tion group studies suggest the proliferation of ergodicgrains or resonating clusters in the system near thetransition [48–50]; furthermore, Refs. [51, 52] suggestthat the inclusion of ergodic grains destabilizes theMBL phase when their length exceeds the critical cor-relation length ξ c = 1 / log(2). In addition, based onthe presence of ergodic grains close to the transition, a r X i v : . [ c ond - m a t . d i s - nn ] M a y Figure 1. Example of colliding eigenstates in the interval 4 . < W < . L = 8. Dotted lines show the values of W where the hybridization is computed, as well as W + dW , for dW = 10 − (exaggerated in distance in the figure tomake it visible). Colors indicate different quantum number configurations. The insets show the hybridization strength | A | between the red and other eigenstates whose range from the red eigenstate is 1 (green and yellow; see pseudo-spinshighlighted in the legend) and 3 (orange and blue); note that the amplitudes of the adiabatically evolved wave functionare obtained by multiplying these values by dW . Notice this differs from the Hamming distance. Notice also that stronglycoupled eigenstates range 1 away from the red eigenstate are O (1) distant in energy, since they are easily coupled throughthe term σ zi σ zi +1 in Eq. (2) through a single (pseudo-)spin flip. On the contrary, eigenstates a larger range away hybridizeonly when close in energy to the red eigenstate. the theory of Refs. [53, 54] predicts the emergence ofa length scale that diverges at the transition.In this work, we consider the evolution of the eigen-states of H ( W ) as we tune the disorder strength W from the MBL phase at large disorder, to the tran-sition, and into the ergodic phase at small disor-der. During this evolution, driven by ‘collisions’, eacheigenstate hybridizes onto the other eigenstates of thesystem (see Fig. 1 for an example), changing theirproperties and strongly influencing the distinct levelstatistics of the MBL and ergodic phases. This hy-bridization occurs over a set of spins spanning a cer-tain size or range R . For strongly hybridized eigen-states there is universal behavior for these collisions.The key results of this work are as follows. • At the transition, we find the probability thattwo eigenstates hybridize over a set of spins is range invariant (Fig. 2 (middle-left)) losing lo-cality in a qualitatively different way from theergodic phase (where hybridization over all clus-ters of spins is equally likely); this suggests theformation of singlets at the transition at alllength scales. • In the MBL phase, the probability of hybridiza-tion decays exponentially with range and weidentify a diverging length scale | α ( W ) | − (seeEq. (6)) from this exponential decay. Addition-ally, hybridization deep in the MBL phase isdominated by sporadic pair-wise collisions. • We identify, and specify the functional formof, a non-diverging correlation length ξ ( W ) (Eq. (7)), which takes the expected value ξ c =1 / log(2) [51, 52] at the transition and closelyagrees, throughout the MBL phase, with corre-lation lengths computed in other capacities [47]. Adiabatic evolution of | n ( W ) (cid:105) and range R hybridization.— Given a fixed disorder realization, i.e. , a set of randomly sampled magnetic fields { h i } ,the model of Eq. (2) defines a Hamiltonian H ( W ) withdisorder strength W . When varying W adiabatically(note that this is different from the local, adiabaticperturbation of Ref. [55]), the eigenstates of H ( W )evolve continuously and we can label them by theirposition in the energy spectrum. Alternatively, we canlabel these eigenstates by their set of quantum num-bers, | n (cid:105) = | ↑↓↓ . . . (cid:105) τ , where the subscript τ indicatesthat this is a product state in the diagonal basis of (cid:96) -bits. In this work, we make use of this labelling byquantum numbers.Let us start by defining hybridization and the con-cept of hybridization rate (see Fig. 1 for an example).Given a differential change W → W + dW , we in-duce the trajectory | n ( W ) (cid:105) → | n ( W + dW ) (cid:105) , in whichthe eigenstate slightly rotates in Hilbert space. In thebasis of eigenstates {| k ( W ) (cid:105)} of H ( W ) we can nowwrite | n ( W + dW ) (cid:105) = c nn | n ( W ) (cid:105) + dim( H ) (cid:88) k (cid:54) = n A nk dW | k ( W ) (cid:105) (3)where, by normalization, c nn = 1 − (cid:113)(cid:80) dim k (cid:54) = n ( A nk dW ) .We call A nk , with k (cid:54) = n , the hybridization rate of | n ( W ) (cid:105) with | k ( W ) (cid:105) . Throughout this paper, we P r ( R ;| A | ) P r ( R ;| A | ) W =0.5 R =1 R =2 R =3 R =4 R =5 R =6 R =7 R =8 R =9 R =10 R =11 P r ( R ;| A | ) P r ( R ;| A | ) W =3.5 R P r ( R ;| A | ) | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A | P r ( R ;| A | ) W =8.0 Figure 2. Probability
P r ( R, | A | ) that two eigenstates hy-bridize over any cluster of range R with hybridization rate | A | , for all pairs of eigenstates over 1024 different disorderrealizations on a chain of size L = 12. We can observethree distinct behaviors. In the ergodic phase ( W = 0 . W = 3 . W = 8 . use dW = 10 − . Note that ( A nk dW ) can also bethought of as a transition probability from | n ( W ) (cid:105) to | k ( W ) (cid:105) . We will be concerned with the probability ofhybridization at various strengths of | A | (note we havedropped indices n and k for simplicity) and ranges R , P r ( R ; | A | ) (see Appendix A for details on the compu-tation of this distribution numerically).We now define the spatial range R associated witha pair of eigenstates | n ( W ) (cid:105) and | k ( W ) (cid:105) . To do so,we look at the difference in their quantum numbers, i.e. their eigenvalues with respect to the (cid:96) -bit oper-ators { τ zi } . These (cid:96) -bits are determined via the di-agonalizing unitary U by Eq. 1. Note that there aremany unitaries which diagonalize H which differ bypermuting the columns of U , as well as changing theirsigns. Making the right choice of U is essential to con-struct the most localized possible set of (cid:96) -bits. Whilefinding the optimal U is not efficiently solvable (andindeed depends on metric of localization chosen), theWegner-Wilson flow approach described in Ref. [56]gives good results in practice. In the present work, wealways refer to the set of operators { τ zi } and to theunitary U found by this method.We say that two eigenstates of a spin Hamiltonian H , | n (cid:105) and | k (cid:105) , have a difference of range R ifthe left-most and the right-most disagreeing quan-tum numbers between the two are R pseudo-sitesaway from each other. For example, eigenstates | n ( W ) (cid:105) = | ↑↓↓↑↓↑(cid:105) τ and | k ( W ) (cid:105) = | ↑↑↑↓↓↓(cid:105) τ differon a cluster of range R = 4 (between the secondand the sixth pseudo-spins). While pseudo-spinsare not exactly spins on real space, they are locallydressed spins in the MBL phase, and so the notionsof distance in real space and in the space induced byl-bits are related. Results.—
When two eigenstates collide, theirdisagreeing pseudo-spins hybridize, while all otherpseudo-spins which are common to both states remainapproximately frozen in their configuration. This hy-bridization drives the eigesntates towards the forma-tion of a cat state across a region of range R . We areinterested in two related probabilities: P r ( R, | A | ) isthe joint probability that a pair of eigenstates | n ( W ) (cid:105) and | k ( W ) (cid:105) hybridize over any cluster of range R witha hybridization rate of magnitude | A nk | . C ( R ; | A | ) isthe probability that a particular cluster of range R hybridizes with magnitude | A | . Note that in the ther-modynamic limit C ( R ; | A | ) = P r ( R ; | A | ) / R , (4) i.e. , the probability that a particular range- R clusterhybridizes goes as the probability any range- R clusterhybridizes over the number of clusters with range R (see Appendix E).Fig. 2 (left column) shows slices of P r ( R, | A | ) as afunction of range R at constant (strong) hybridizationrates | A | . It can be seen that this probability scalesexponentially with range R for all W as P r ( R ; | A | ) ∝ exp [ α ( W ) R ] (5)The slope of the semi-log plots in Fig. 2 (left), α ( W ),is shown in Fig. 3 (top).In the ergodic phase, α >
0; furthermore, deep inthe ergodic phase
P r ( R ; | A | ) is proportional to thenumber of clusters of range R (see dashed lines ofFig. 3 (top)), which scales as 2 R in the thermody-namic limit, and slower than that in finite systems(see Appendix E). This suggests the notion of localityis completely lost in the ergodic phase, and the prob-ability, C ( R ; | A | ), that a particular cluster of pseudo-spins hybridizes is constant, independent of its range R .In the MBL phase and around the transition α ( W )scales as α ( W ) = − B log( W/W c ), (6)as seen in Fig. 3 (bottom left). B is a constant (shownin the inset) and numerically trends toward 1 / W c is the critical point identified by where α ( W c ) = 0;note the logarithmic fit breaks deep in the ergodicphase. The critical point W c trends upward as afunction of L , reaching a thermodynamic value above W L W c L = 8 L = 10 L = 12 L W c ( L ) W = B log ( WW c ) W L =32 (OPO, W c =3.7) = B log ( BWW c ) c = L B Figure 3.
Top:
Exponent α ( W ) of the scaling of P r ( R ; | A | ) ∝ exp( αR ) at constant | A | (see left panel ofFig. 2) for systems of size L = 8 , ,
12 computed by aver-aging the slope obtained from a linear fit of log(
P r ( R ; | A | ))as a function of R , for all values of | A | in the interval[10 − . , − . ] ([10 − , − . ] for L = 8); errorbars rep-resent their standard deviation. Critical points W c ( L )(stars and inset) defined by α ( W c ) = 0 (range invariance in P r ( R, | A | )). Dashed lines show the expected α if the prob-ability of hybridization between any two spin configura-tions were constant (see Appendix E). This goes to log(2)(top red line) in the thermodynamic limit. Bottom left:
Fit of the form α = − B log[ W/W c ( L )] for W > W c ( L )with inset showing values of B as a function of 1 /L ; red-dashed line is B = 1 /
2. The fit breaks down deep in theergodic phase.
Bottom right:
Black lines show the lo-calization length ξ ( W ) of Eq. (7) for L = 8 , , ,
32; for L = 32 we use W c = 3 . B = 1 /
2. Brown squaresshow the correlation length of one-particle orbitals (OPO)as reported in Ref. [47] (Fig. 4, L = 32, energy density (cid:15) = 0 . ξ ( W ) = [ − α ( W ) − + log(2)] − for L = 8 , ,
12. Vertical dashed lines show the value of W c for each system size; ξ = 1 / log(2) at W = W c . All pan-els:
We have tested averaging various different spans of R and | A | . The logarithmic dependence of α for W ≥ W c appears very robust to all choices; the particular valuesof W c and β for a given L can change at the 20% leveldepending on the choice of fit. W ≥ .
25. As expected for a localized state, α < any clus-ter ( P r ( R ; | A | )) of range R hybridizes is exponentiallysuppressed with R . Notice that, in the MBL phase,this gives an algebraic dependence with W which goesas P r ( R, | A | ) ∝ ( W/W c ) − BR .Strikingly, at the transition P r ( R, | A | ) becomes range-invariant at strong values of | A | out to a range-dependent cutoff in | A | after which P r ( R, | A | ) dropsthe universal, range-independent behavior (see rightcolumn of Fig. 2); the larger R , the larger the cut-off. Notice this is consistent with Eq. (6) at W = W c ( α ( W c ) = 0) and allows us to identify a length scale η ( W ) = [ − α ( W )] − for W ≥ W c that diverges as W → W c . It should be emphasized that this isa special type of ‘cluster-agnostic range-invariance’which differs qualitatively from the sort of range-invariance found in the ergodic phase. At the transi-tion, P r ( R ; | A | ) is constant, while in the ergodic phase C ( R ; | A | ) is constant.It is also possible to define a correlation length ξ ( W )from C ( R ; | A | ) = exp[ − R/ξ ( W )] at large, fixed | A | .Working in the thermodynamic limit, we have that ξ ( W ) − = η ( W ) − + ξ − c = (cid:20) B log (cid:18) /B WW c (cid:19)(cid:21) − (7)where the second equation follows from our numer-ically determined functional form for α ( W ). Thecritical correlation length ξ c = ξ ( W c ) = log(2) − is in agreement with the value that Refs. [51, 52]find to destabilize the MBL phase Note that in ourcase the value of ξ c comes from positing ‘cluster-agnostic range-invariance’ at the critical point (where C ( R ; | A | ) becomes independent of R ), and is indepen-dent of the functional form of α ( W ) and requires noadditional assumptions about ergodic grains.Using B = 1 / W c = 3 .
7, we compare ξ ( W ) of Eq. (7) with the corre-lation lengths found in Ref. [47], which were computedfrom the decay of approximate (cid:96) -bits generated from L = 32 MBL eigenstates. The comparison is shownin Fig. 3 (bottom right) and the agreement is striking. Pair-wise collisions in MBL.—
We now showthat, in the MBL phase, the probability of colliding (atfixed R ) with a given strength | A | (hybridization rate)decays as a power law with strength P r ( | A | ; R ) ∝| A | − / . This algebraic decay comes from the avoidedlevel crossings of two eigenstates, suggesting that hy-bridization in the MBL phase is dominated by spo-radic pair-wise collisions (note this is related to theresonating pairs of configurations found in Ref. [57]to be responsible for the conductivity deep in theMBL phase as well as the rare resonances consideredof Ref. [58]). On the contrary, ergodic eigenstatesare continuously hybridizing through overlapping pro-cesses, which are due to continuous collisions.We compute the exponent β of the universal powerlaw decay of P r ( R, | A | ) ∝ | A | β , for constant (large) R ,right before P r ( R, | A | ) hits the cutoff characteristic ofeach range R (see Fig. 2 (right) and Appendix G).The top panel of Fig. 4 shows the values of β asa function of W and R for different system sizes L . At large W and R , we find β → − . Theexponent β = − can be obtained from a toytwo-level model H toy ( λ ) ≡ H + λH for which W R - . - . - . - . L = 8 W - . - . - . - . - . L = 10 W - . - . - . - . L = 12
100 50 0 50 10015010050050100150 E | A | P r ( | A | ) = 1.50 Figure 4.
Top: colormap with the exponent β of thepower law decay of P r ( R, | A | ) ∝ | A | β at strong | A | fordifferent values of R and W . We can see that β → − atlarge R and strong W . Bottom: level collision of the two-level toy model H toy ( λ ) = H + λH and its corresponding P r ( | A | ). We recover the exponent β = − , which suggeststhat strong, long-range hybridization in MBL is dominatedby rare, pair-wise collisions. two states | λ ) (cid:105) and | λ ) (cid:105) undergo a collision.As can be seen in the bottom panel of Fig. 4, bysampling hybridization rates between | (cid:105) and | (cid:105) , | A | = |(cid:104) λ ) | λ + dλ ) (cid:105)| uniformly at random over alarge window in λ ([ − λ m , λ m ≡ P r ( | A | ) that decaysas a power law with exponent β = − . This behaviormimics the rare pair-wise collisions of the MBL phase. Conclusions.—
In this work, we have consideredthe hybridization of eigenstates as they are drivenfrom the MBL phase, through the transition, and intothe ergodic phase. In particular, we have studiedtwo related probabilities:
P r ( R ; | A | ) and C ( R ; | A | ). P r ( R ; | A | ) is the probability that a pair of eigenstateshybridize over a region of range R (for fixed and stronghybridization rate or strength | A | ), while C ( R ; | A | )is the probability that they hybridize over a specificcluster of pseudo-spins spanning a region of range R .While both probabilities are suppressed exponentiallywith R in the MBL phase, P r ( R ; | A | ) is range invari-ant at the transition, and C ( R ; | A | ) is range invariantdeep in the ergodic phase. This lets us define twolength scales: η ( W ) and ξ ( W ). η ( W ) is related to P r ( R ; | A | ) and represents a length scale that divergesat the transition. ξ ( W ) is related to C ( R ; | A | ); it hasthe anticipated correlation length 1 / log 2 at the tran-sition and is a surprisingly close match to other corre-lation lengths computed in the MBL phase. We iden-tify a functional form for P r ( R ; | A | ) in the MBL phaseand around the transition, which can be extended tofunctional forms for C ( R ; | A | ), η and ξ . We anticipatethat the ansatz of Eq. (7) will be key in understanding the phenomenology of the MBL transitions.The phenomenology of the transition is oftenthought of as being driven by the proliferation of res-onances or cat states which are believed to be re-sponsible for melting the MBL phase. Our findingof range-invariant hybridization (at all large enoughhybridization strengths) of eigenstates at the transi-tion suggests the proliferation of singlets - cat states- at all scales, which are related to the appearanceof resonances in a range-invariant fashion across thesystem. To identify the range-invariant hybridizationand correlation lengths required us both to work in thepseudo-spin basis, abandoning strict locality in favorof a quasi-local view of the system, as well as sepa-rating out probability distributions by hybridizationstrengths so range-dependent cutoffs do not contami-nate averages.It is natural to expect locality to break down atthe MBL-ergodic transition. There are two reasonableways this could happen: either C ( R ; | A | ) or P r ( R ; | A | )become range-invariant. Our observations find thatthe transition corresponds to the latter, which aloneis sufficient to set the value of ξ c = 1 / log(2) found inthe literature, without assumptions on the inclusionof ergodic grains.Finally, Eq. 7 suggests the intriguing fact that thecritical point W c could be determined by properties ofthe system deep in the MBL phase; it also suggests anadditional way to interpret current correlation lengthsin the literature. We also note that P r ( R ; | A | ) scalesalgebraically with W and it is an interesting questionwhether this has any relationship to the algebraic scal-ing with W seen in Ref. [59] for the smallest coupledenergy scales within the MBL phase. ACKNOWLEDGMENTS
The authors would like to thank Vedika Khemanifor pointing out the importance of clusters with morethan two resonating sites as a signature of many-bodyeffects. We are grateful to David Luitz and GregHamilton for carefully reading and providing com-ments on our manuscript. We are also thankful toSarang Gopalakrishnan for pointing out the role ofresonances in the conductivity in MBL, as well asproviding a relevant reference. We would like to ac-knowledge useful discussions with Greg Hamilton, EliChertkov, David Huse, Romain Vasseur, Chris Lau-mann, and Anushya Chandran. We both acknowledgesupport from the Department of Energy grant DOEde-sc0020165. BV also acknowledges support from theGoogle AI Quantum team. This project is part of theBlue Waters sustained petascale computing project,which is supported by the National Science Founda-tion (awards OCI-0725070 and ACI-1238993) and theState of Illinois. 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M¨uller, V. Khemani, M. Knap,E. Demler, and D. A. Huse, Physical Review B ,104202 (2015).[58] J. A. Kj¨all, Physical Review B , 035163 (2018).[59] X. Yu, D. Pekker, and B. K. Clark, arXiv:1909.11097[cond-mat, physics:quant-ph] (2019), arXiv:1909.11097. [60] V. K. Varma, A. Raj, S. Gopalakrishnan,V. Oganesyan, and D. Pekker, Physical ReviewB , 115136 (2019). P r ( R ;| A | ) P r ( R ;| A | ) W =0.5 R =1 R =2 R =3 R =4 R =5 R =6 R =7 P r ( R ;| A | ) P r ( R ;| A | ) W =3.5 R P r ( R ;| A | ) | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A | P r ( R ;| A | ) W =8.0 Figure 5. Slices of
P r ( R ; | A | ) for L = 8. This is similar toFig.2 of the main text. Appendix A: Obtaining the joint probabilitydistribution
P r ( R ; | A | ) In order to numerically obtain the joint probablilitydistribution
P r ( R ; | A | ) (see Fig. 2 in the main text),we first compute the absolute value of all hybridizationratios between all pairs of eigenstates over 1024 ran-dom realizations of the Hamiltonian in Eq. (2) of themain text. We do this for different disorder strengths W and system sizes L . Each hybridazation ratio isassociated to a range R ∈ [1 , L − L, W, R ) we now have a set of hybridizationratios ( i.e. , a set of values of | A | ).Then, for each ( L, W, R ) we histogram the corre-sponding set with log-spaced bins ( i.e. , with bins thatare equally spaced on a logarithmic scale); this givesus h L,W,R ( | A | i ), where | A | i is a discrete variable thattakes the middle point of bin i . In order to preciselyobtain a probability distribution, we now need to di-vide each h L,W,R ( | A | i ) by the size of bin i ; we denotethe result by f L,W,R ( | A | i ). Finally, we enforce normal-ization for each tuple ( L, W ) by dividing f L,W,R ( | A | i )by S L,W ≡ (cid:80) R (cid:80) i f L,W,R ( | A | i ), after which we ob-tain P r ( R ; | A | i ) ≡ f L,W,R /S L,W , which for simplicitywe just denote by
P r ( R ; | A | ). Appendix B:
P r ( R ; | A | ) for L = 8 , In Figs. 5 and 6 we show the equivalent of Fig. 2 ofthe main text for L = 8 , P r ( R ;| A | ) P r ( R ;| A | ) W =0.5 R =1 R =2 R =3 R =4 R =5 R =6 R =7 R =8 R =9 P r ( R ;| A | ) P r ( R ;| A | ) W =3.5 R P r ( R ;| A | ) | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A |=10 | A | P r ( R ;| A | ) W =8.0 Figure 6. Same as Fig. 5 for L = 10. Appendix C: Alternative view of the collapse of
P r ( R ; | A | ) at large | A | to a universal function atthe transition The probability distribution
P r ( R ; | A | ) collapses toa universal function at large enough | A | at the ergodic-MBL transition. This is difficult to appreciate clearlyfrom Fig. 2 of the main text. In Fig. 7 we plot slices of P r ( R ; | A | ) | A | at fixed R , which simply shifts the firstderivative of the curves (in a log-log scale) by 3. Thisview allows us to appreciate the collapse at the tran-sition (see W = 3 . L = 12) of the curves for dif-ferent ranges onto a universal functional form at largeenough | A | and before they hit their range-dependentcutoff. Note that the curves are modified equally forall ranges, and so a collapse of P r ( R ; | A | ) | A | impliesautomatically a collapse of P r ( R ; | A | ). Appendix D: Alternative visualization of
P r ( R ; | A | ) Here we present an alternative visualization of thejoint probability distribution
P r ( R ; | A | ). Instead ofshowing slices at constant R or | A | , we show a col-ormap of the distribution with contour lines (Figs. 8,9, and 10). We can see that at large | A | the probabil-ity that a pair of eigenstates hybridize over a partic-ular range becomes range invariant at the transition,with contour lines becoming vertical (see the case of L = 12, W = 3 .
5, over the shaded area). We canalso appreciate finite size effects at the largest R foreach L , as well as R -dependent cutoffs for the smallest P r ( R ;| A | ) | A | W = 0.5 L = 8 L = 10 L = 12 R =1 R =2 R =3 R =4 R =5 R =6 R =7 R =8 R =9 R =10 R =11 P r ( R ;| A | ) | A | W = 3.5 | A | P r ( R ;| A | ) | A | W = 8.0 | A | | A | Figure 7. Slices of fixed R of P r ( R ; | A | ) | A | as a function of | A | for different system sizes. We can appreciate a collapseof the curves onto a universal functional form at the transition W ≈ . | A | and before each curve hitsits R -dependent cutoff. ranges, where P r ( R ; | A | ) drops its universal behavior.Finally, equally spaced contour lines imply a powerlaw in P r ( R ; | A | ) ∝ | A | b eta ) at constant R , as dis-cussed in the main text.The values of α presented in the main text were ob-tained from fits (log[ P r ( R ; | A | )] = αR + const. ) overslices at constant | A | over the shaded areas, i.e. , avoid-ing finite size effects and constraining the fits to large | A | , while avoiding R -dependent cutoffs. Appendix E: Combinatorial counting of clustersas a function of range
Given two eigenstates and their respective sets ofpseudo-spins, they can disagree over a subset or clus- terthat spans a region of range R . Here we derive ananalytical formula for the number of possible clustersfor a fixed R , N ( R ).First, note that two sets of quantum numbers haveto disagree in an even number of sites, given the factthat we are working on the zero magnetization sec-tor. Furthermore, the magnetization of the cluster ofdisagreeing quantum numbers must also be zero, i.e. ,have an equal number up and down pseudo-spins. Wecan now proceed with the counting.For a fixed range R , clusters can take any order o = 2 , , , . . . R (number of pseudo-sites). For eachorder, and ignoring cyclic permutations of the bit-strings, two of the disagreeing pseudo-spins have tobe “pinned” at the left and right ends of the range R interval, leaving only freedom to the inner o − (cid:0) R − o − (cid:1) ways to place them.This yields: R (cid:88) even o (cid:18) R − o − (cid:19) (E1)ways to place disagreeing pseudo-spins while forminga range R cluster. There is another factor to takeinto account: once the o disagreeing pseudo-spins areplaced, their orientations have to be chosen. Giventhat the magnetization of the o pseudo-spins is zero,there are (cid:0) oo/ (cid:1) ways to arrange them, times the num-ber of ways the agreeing pseudo-spins can be arranged,which is (cid:0) L − o ( L − o ) / (cid:1) , and where L is the total number ofsites in the system. We then get that the scaling is: ∝ R (cid:88) even o (cid:18) R − o − (cid:19)(cid:18) oo/ (cid:19)(cid:18) L − o ( L − o ) / (cid:19) . (E2)Finally, we have to account for tranlations of thebitstrings on the chain. Given open boundary condi-tions, the number of cyclic permutations allowed fora particular cluster of range R is equal to L − − R .We finally have the right scaling: N L ( R ) = R (cid:88) even o (cid:18) R − o − (cid:19)(cid:18) oo/ (cid:19)(cid:18) L − o ( L − o ) / (cid:19) ( L − − R ) . (E3)In the thermodynamic limit, this goes as ∝ R . Thecurve N L ( R ) was plotted in Fig. 2 (top-left, dashedlines) for convenient comparison with P r ( R ; | A | ) as afunction of R deep in the ergodic phase. While thevalue of α (see main text) is log(2) in the thermody-namic limit, for finite size systems it can be computednumerically from Eq. (E3). Appendix F: Relation of ξ to other correlationlengths While there are a number of other correlationlengths in the literature which scale as log( W ), notall of them exhibit B = 1 /
2; note that log( W ) is thescaling expected for the correlation length of an An-derson insulator at strong disorder. Since rescalingour units of length multiplicatively by s rescales ourinverse correlation length as 1 /s , it is a reasonable con-jecture that we could fit other correlation lengths as ξ − ( W ) = − B/s log(4
W/W c ). Using this ansatz, wetest our results against two other correlation lengths.These are not chosen generically but are selected tobe ones which both show the expected logarithmicscaling and have small errors. First, we considerRef. [60] (IOM ( xX ) from their Fig. 1); this corre-lation length scales logarithmically but with B/s ≈ W c of 4.68. Finally, we consider the freezing exponent of the recent preprint,Ref. [46], which is inversely proportional to the corre-lation length and yields a critical value of W c ≈ . B/s ≈ .
67. This is consistent with an inde-pendent estimate of the W c in their paper. Appendix G: All β fits Here we provide all linear fits reported in Fig. 4 ofthe main text for the computation of the exponent β . Fig. 11 reports the all fits for L = 8, Fig. 12 re-ports L = 10 fits, and Fig. 13 reports L = 12 fits.We extract β for a linear fit in the log-log plot of P r ( R ; | A | ) as a function of | A | for fixed R . In orderto decrease noise in the fit, we perform fits over sev-eral intervals around a central interval in | A | (shownby the two red vertical lines); we then average theresults. In all cases, the background of the plots fol-lows the same colormap as in Fig. 4 of the main text.Ranges R = 1 , R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 2.0 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 4.0 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 5.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 6.0 | A | R . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 6.5 | A | . . . . . . . . . . . . . . . . . . . . . . . . . . W = 7.0 | A | . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 7.5 | A | . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 8.0 Pr ( R ; | A |) Figure 8. Colormap of
P r ( R ; | A | ) with contour lines for L = 8. For visibility in the plots, the values on the contourlines correspond to log [ P r ( R ; | A | )]. The exponent α was computed from exponential fits to slices of P r ( R ; | A | ) on theshaded area. R - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 0.5 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 1.0 - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 1.5 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 2.0 R - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 2.5 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 3.0 - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 3.5 - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 4.0 R - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 4.5 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 5.0 - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 5.5 - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 6.0 | A | R - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 6.5 | A | - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 7.0 | A | - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 7.5 | A | - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 8.0 Pr ( R ; | A |) Figure 9. Same as Fig. 8 for L = 10. R - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 0.5 - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 1.0 - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 1.5 - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 2.0 R - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 2.5 - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 3.0 - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 3.5 - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 4.0 R - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 4.5 - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 5.0 - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 5.5 - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 6.0 | A | R - . - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 6.5 | A | - . - . - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 7.0 | A | - . - . - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 7.5 | A | - . - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = 8.0 Pr ( R ; | A |) Figure 10. Same as Fig. 8 for L = 12. P r ( R ;| A | ) WR P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) P r ( R ;| A | ) | A | P r ( R ;| A | ) | A | | A | | A | | A | | A | | A | Figure 11. Empirical fits of