Characterizing the many-body localization transition through correlations
CCharacterizing the many-body localization transition through correlations
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: July 15, 2020)Closed, interacting, quantum systems have the potential to transition to a many-body localized(MBL) phase under the presence of sufficiently strong disorder, hence breaking ergodicity and failingto thermalize. In this work we study the distribution of correlations throughout the ergodic-MBLphase diagram. We find the typical correlations in the MBL phase decay as a stretched exponentialwith range r eventually crossing over to an exponential decay deep in the MBL phase. At thetransition, the stretched exponential goes as e − A √ r , a decay that is reminiscent of the randomsinglet phase. While the standard deviation of the log( QMI ) has a range dependence, the log(
QMI )converges to a range-invariant distribution on all other moments ( i.e. , the skewness and higher) atthe transition. The universal nature of these distributions provides distinct phenomenology of thetransition different from both the ergodic and MBL phenomenologies. In addition to the typicalcorrelations, we study the extreme correlations in the system, finding that the probability of stronglong-range correlations is maximal at the transition, suggesting the proliferation of resonances there.Finally, we analyze the probability that a single bit of information is shared across two halves of asystem, finding that this probability is non-zero deep in the MBL phase but vanishes at moderatedisorder well above the transition.
PACS numbers: 75.10.Pq,03.65.Ud,71.30.+h
I. INTRODUCTION
While most phases of matter are related to the prop-erties of the ground state or the thermal density ma-trix of a system, eigenstate phases of matter are char-acterized by properties of a system’s interior eigen-states. The two most well-known eigenstate phasesare the many-body localized (MBL) phase, presentin sufficiently disordered interacting systems, and thestandard ergodic phase into which it transitions [1–6].The eigenstates of the MBL and ergodic phases arequalitatively different in their properties, which affectsthe dynamical properties of their system. MBL eigen-states show area-law entanglement [7–10], their corre-lations typically decay quickly with range [11], and lo-cal observables vary wildly with energy, thus violatingthe eigenstate thermalization hypothesis (ETH) [4,12–16]. On the contrary, ergodic eigenstates showvolume-law entanglement, correlations typically decayslowly, and the ETH is satisfied as local observablesvary smoothly with energy. While much is knownabout eigenstates in the MBL and ergodic phases, theproperties of eigenstates at the critical point betweenthese phases are less well understood. So far, the plu-rality of numerical evidence suggests localized eigen-states with sub-volume law entanglement [6, 10, 17–19], bimodality in the distribution of the entanglemententropy [9], and some forms of range-invariance [20–23]. In addition, there is a body of work on renormal-ization group (RG) approaches to the phenomenologyof the MBL transition [24–33]. Transitions in Floquetmodels have also been considered, with average long- ∗ [email protected] † [email protected] (independent of L) C ≈ 0.38 Typical correlations: exp[− Ar β ] β ≈ 1 Maximal range-invariant probability of QMI ≥ log(2) β = 1/2 skewness (and higher moments) of log(QMI) range invariant μ ( r ) finite p ( λ → log(2)) of log(QMI): σ Cr + D p ( λ → log(2)) = 0 Figure 1. Summary of results of this work on the distribu-tion of log(
QMI ) and the second Schmidt eigenvalue ( λ )of the half-cut reduced density matrix. Blue bar showsthe region of stretched exponential decay as a functionof range r (exp (cid:2) − Ar β (cid:3) ) of the typical QMI (equivalentlypolynomial decay of the mean of log(
QMI )) starting at β = 1 / W / = 2 . β ≈ W ∼ QMI ) (yellow) is linear atall W in the MBL phase with an L -independent cross-ing of C as marked on the figure. The range-invarianceof the skewness (and higher moments) of log( QMI ) alsohappens at W = W / = 2 .
9. The probability of findingrange-invariant strong values of the
QMI ≥ log 2 is largestat the W shown, suggesting the proliferation of multi-siteresonances. The green bar indicates the region where theprobability of λ = log 2 is finite. All values are at finite L = 18 and we expect that 2 . < W < L , and likely trendtowards the same value of W in the thermodynamic limit. distance correlations peaking at the transition show-ing system size independence [34].In this work we focus on the correlations across asystem throughout the MBL-ergodic phase diagram,providing extensive phenomenology in the MBL phaseand at the transition. Correlations have played keyroles in understanding phases of matter and critical a r X i v : . [ c ond - m a t . d i s - nn ] J u l r l o g ( Q M I ) QMI =2log(2)Machineprecision
QMI typ = const . QMI typ = e Ar QMI typ = e Ar W = 1.0 W = 3.5 W = 8.0MeanMedian Figure 2. Probability distributions of the logarithm of thetwo-site
QMI for a system of size L = 18 for all ranges r = | i − j | (between sites i and j ) in the ergodic phase( W = 1), around the transition ( W = 3), and deep inthe MBL phase ( W = 8). Stars indicate the mean of thedistributions and triangles indicate their median, whichshows similar behavior. While the typical (log-averaged) QMI is constant with range in the ergodic phase, it decaysexponentially deep in the MBL phase, and as a stretchedexponential of the form e − Ar β , with 1 / < β <
1, atmoderate values the of disorder strength on the MBL sideof the phase diagram. At the transition, the decay followsa stretched exponential with β = 1 / i.e. , QMI typ = e − A √ r . We only consider distributions that are well abovemachine precision, i.e. , those with at least 99% of theirmass above QMI = 10 − (Appendix B). points. In disordered systems, understanding the dis-tribution of correlations, including their typical val-ues, has been particularly insightful. One canonicalexample of this is the random singlet phase, whichappears as a universal fixed point in the strong disor-der renormalization group (SDRG) analysis of manydisordered ground state spin systems [35–39]. Inthe random singlet phase, typical correlations exhibitstretched exponential behavior and universal featuresare anticipated for the full distribution of correla-tions [35].One way to quantify correlations is through thequantum mutual information (QMI). The primarytool of this paper is the computation of the QMI in thespin- nearest-neighbor antiferromagnetic Heisenbergchain with random onsite 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 . (1)where the onsite magnetic fields { h i } are sampled uni-formly at random from [ − ,
1] and W is the disorderstrength. The model of Eq. (1) has been studied ex-tensively in the context of MBL [6, 8, 9, 11, 16, 19,22, 40–54]. The two-site QMI was introduced in thecontext of MBL in Ref. [11], where the authors foundevidence for exponentially decaying QM I with range in the MBL phase and slower decay in the ergodicphase. Our goal will be to look at the distributionsof the QMI considering both the typical and extreme(atypically strong) correlations.The key results of this work are the discovery of • Stretched exponential behavior, exp (cid:2) − Ar β (cid:3) (where r is the range between two spins), oftypical correlations both at the transition andin the MBL phase, spanning from β = 1 / β = 1 around W ≈
8. Interestingly, the random singlet phasehas the same decay of the typical correlations asthe MBL-ergodic transition. • Range-invariant universal (in the skewness andhigher statistical moments) distributions of thelog(
QM I ) at the transition. Even excess stan-dard moments of these distributions are zero. • Range-invariant strong pairwise
QM I at thetransition suggesting the existence of resonatingcat states at all ranges at the critical disorderstrength between the MBL and ergodic phases.Note the idea of a stretched exponential scaling ofvarious quantities at moderate disorder has appearedin the MBL literature. Refs. [27, 29, 55] discuss astretched exponential decay of the average (not typi-cal) correlations in the MBL phase from a simplifiedRG analysis, a microscopically motivated RG scheme,and a toy model, respectively. Refs. [31, 32] considerinstead the size of ergodic inclusions in the system;through RG arguments they find stretched exponen-tial scaling of the size of these inclusions in the MBLphase and power law decay of their size at the transi-tion. Using a heuristic numerical algorithm, Ref. [53]finds evidence for the algebraic scaling of the clustersizes at the transition, crossing over to a stretched ex-ponential scaling of the cluster sizes upon entering theMBL phase and eventually becoming exponential atstrong disorder strengths.Our results are qualitatively different, findingstrong numerical evidence for stretched exponentialbehavior of typical correlations both in the MBLphase and at the transition. Interestingly, our numer-ics are cleanest and most compelling at the criticalpoint. We note the stretched exponential behaviorwe find is clearly distinct from a power law. It is aninteresting open question how this compares to the av-erage correlations found in various RG analyses andtoy models as well as how typical correlations relateto the size of ergodic grains.In Section II we analyze the structure of the typicalcorrelations as well as look at the various moments ofthe log ( QM I ). In Section III we show our results onthe extremal values of the
QM I and their relation toscale invariant resonances. In Section IV we discussthe statistics of the second singular value of the bipar-tite entanglement entropy which has been proposedrecently as a robust order parameter in the ergodic-MBL phase diagram [56]. Finally, in Section V wesummarize our findings and discuss their implications.For L = 18, we obtain 100 eigenstates close to en-ergy density (cid:15) ≈ . disorder realizations, obtaining a total of 10 eigen-states. For L = 14 ,
16, we obtain 5 eigenstates closeto (cid:15) ≈ . × disorder realizations, obtaining also a total of10 eigenstates per system size. We do this for differ-ent values of the disorder strength W . II. TYPICAL CORRELATIONS
In this section we look at the typical values of two-point correlations in an eigenstate of the Hamiltonianof Eq. (1) throughout the ergodic-MBL phase dia-gram. We use the
QM I between all pairs of sitesin a one-dimensional spin chain as a measure of thestrength of their correlation that is agnostic to thechoice of any particular correlation function. The
QM I measures all correlations, both classical andquantum, between subregions in a system. The
QM I between subregions A and B is defined as: QM I AB ≡ S A + S B − S AB , (2)where S A is the Von Neumann entanglement entropybetween subsystem A and its surroundings; we alwayswork with the QM I between pairs of sites, i and j ,which we denote QM I ij . The two-site QM I has amaximum value of 2 log(2), which occurs when twosites form a singlet. However, in many-body systemsit is very rare for two sites to form a singlet with-out being entangled to other sites; in the case of amulti-site singlet (i.e. linear superposition betweentwo product states which differ on k >
QM I between two sites is equal to log(2). We define r as the range between two sites, i.e. , r ≡ | i − j | .Ref. [11] finds that the typical values of the QM I de-cay exponentially with r in the MBL phase and slowerthan exponentially in the ergodic phase. Here we fo-cus in detail on the question of the behavior of thetypical correlations along a one-dimensional systemin the ergodic-MBL phase diagram.We work with the distributions of the log( QM I )(see Fig. 2, where, for readability, the log ( QM I )is presented), as opposed to the distributions of the
QM I . We consider the log(
QM I ) for each range r separately. A first visual inspection shows compactdistributions that are constant across ranges at weakdisorder and decaying and broadening (with r ) distri-butions at moderate and large disorder. Also, the dis-tributions seem skewed in opposite directions at largeand small disorder strengths.In Section II A we study the decay of the typicalcorrelations with r ; surprisingly, we find a region inthe MBL side of the phase diagram with a stretchedexponential decay at moderate values of the disor-der strength W terminating at the transition witha stretched exponential with exponent 1 /
2; this hassimilarities with the random singlet phase that arisesas a fixed point in renormalization group studies of l o g ( Q M I t y p ) W = 0.5 = 0.00 W = 1.5 = 0.06 r l o g ( Q M I t y p ) W = 3.0 = 0.54 r W = 8.0 = 0.98 Figure 3. Log-log plot of − log ( QMI typ ) as a function of r for a system of size L = 18. We can see that for moderateand large disorder strength ( W = 3 . . QMI typ = e − Ar β ) fitswell the data at large r . A linear fit to the curves is shown,as well as the interval of data taken for the fit (red verticallines). Deep in the ergodic phase ( W = 0 . QMI typ isconstant. At slightly higher values of the disorder strength( W = 1 . L and W . disordered systems [35, 38, 39]. In Section II B welook at the standard deviation of these distributions,finding they increase linearly with range r . Next, inSection II C, we study the skewness and higher sta-tistical moments of the distributions; our results showthat these moments take a universal value at the tran-sition for large enough ranges. This implies that thedistribution of log( QM I ) is universal at the transitionbeyond the first two moments. Finally, in Section II Dwe summarize and discuss our findings on the typicalcorrelations. As we can see in Fig. 2, the QMI reachesmachine precision ( ≈ − ) at large range r and largedisorder strength W ; we only consider those points( i.e. the triplet ( L, W, r )) for which the distributionof the log(
QM I ) has at least 99% of its mass above
QM I = 10 − , i.e. , one order of magnitude abovethe machine precision threshold of double-precisionfloating-point numbers (see Appendix B). A. The decay of QMI typ
The typical values of the
QM I are defined as thelog-averaged
QM I : QM I typ ≡ (cid:104)
QM I (cid:105) log = e (cid:104) log( QMI ) (cid:105) , (3) i.e. , it is computed by exponentiating the mean of thedistributions of Fig. 2. We find that QM I typ fits astretched exponential of the form
QM I typ = e − Ar β , (4) W W L = 14 L = 16 L = 18 L W / L = 14 A v e r a g e r e s i d u a l s q u a r e d p e r p o i n t i n f i t L = 16 B W L = 18 W W A W A min W L = 14 L = 16 L = 18 L W A m i n Figure 4.
Top:
Exponent β of the stretched exponentialof the decay of the typical QMI , QMI typ = e − Ar β . In-set shows W / (at which β = 1 /
2) as a function of 1 /L ( W / = 2 . , . , .
88 at L = 14 , ,
18, respectively).
Middle left:
Average residual squared per point in thelinear fit
QMI /βtyp = − Ar + B . Middle right: B as afunction of W . B is zero if the stretched exponential as-sumption was correct ( W > W / ). Bottom: coefficient A as a function of W . Inset shows W A min as a functionof 1 /L. Middle and bottom:
We show dashed verticallines at the values of W / extracted from the curves inthe top panel. at large range r in the MBL phase and at the tran-sition. This is demonstrated by the linear behav-ior on the log-log plot of − log ( QM I typ ) in Fig. 3.This linear fit is especially compelling at the transi-tion ( W ≈ . L = 18) where essentially all rangesare well fit by a linear curve. In the ergodic phase, atintermediate values of the disorder strength ( W = 1 . W = 0 .
5) we find that
QM I typ is constant with r . We can extract the exponent β from the slope of the fit to the log-log plot. The values of β are pre-sented in the top panel of Fig. 4 (confidence inter-vals are defined by the maximum (minimum) β foundover all linear fits of three or more consecutive pointsin the region fitted; see Appendix E). As discussedabove, the values of β are not reliable at low disor-der strength; despite this, we present all values of β even when not reliable. By visual inspection, we con-sistently find, across different system sizes L , that thefits from which we extract β are of good quality above W / , which we define as the value of W at which β = 1 / . Interestingly, the data on the log-log plotsfrom which β is extracted falls below the linear fit atlow ranges for weak disorder, while it lays above thelinear fit at large values of W . All ranges, includinglow range data, fall exactly on top of the linear fit pre-cisely when β = 1 / W / up to a value of W for which β ≈
1. The inset of Fig. 4 shows W / as a func-tion of 1 /L for the three values of L we measure. Anaive extrapolation to L → ∞ seems consistent with W / coinciding with the critical value of W in thethermodynamic limit, i.e. , W / ( L → ∞ ) = W c ≈ QM I typ follows a stretched exponential (Eq. 4) downto the value of W for which β = 1 / W / ), wepresent in the middle-left panel of Fig. 4 the aver-age residual squared per point in the fits from which β was extracted, i.e. , log-log plots like those of Fig. 3.We can see that the residuals are consistent with high-quality fits at and above W / , where they are practi-cally zero. Below W / (shaded our region) the resid-uals per point rapidly increase.Finally, the bottom panel of Fig. 4 shows the val-ues of A in the stretched exponential as a function of W for different system sizes L . We extract A fromthe slope of a linear fit of log( QM I typ ) /β = A (cid:48) r + B ,where β takes the empirically obtained value of thetop panel of Fig. 4, and A (cid:48) = A /β . Note that thisis only correct if B = 0 in the fit, which simultane-ously corresponds to our stretched exponential ansatzbeing correct. Indeed, these fits find B to be practi-cally zero (within error bars) for W ≥ W / , as shownin the middle-right panel of Fig. 4, which is an ex-cellent a posteriori consistency check for our ansatz,independent of the residuals of the middle-left panel.On the contrary, B grows rapidly below W / , wherethe ansatz breaks. As in the case of β , we show inthe bottom panel all values of A found, regardlessof their reliability. We have highlighted two sets ofpoints (marked as stars). First, the values of A ( W / )show an increasing trend as W / shift towards highervalues of W with system size; we will revisit this inSection II D. Second, we drive the reader’s attentionto the points at which A is minimal, W A min . The QM I decays exponentially deep in the MBL phase, i.e. , QM I typ = e − Ar ; since the system should local-ize further as W increases, A must increase with W if the decay is exponential. Note 1 /A is a localization r [ l o g ( Q M I )] L = 18 [log( QMI )]= Cr + D W =0.5 W =1.0 W =1.5 W =2.0 W =2.5 W =2.8 W =3.0 W =3.1 W =3.2 W =3.3 W =3.4 W =3.5 W =3.6 W =3.7 W =3.8 W =3.9 W =4.0 W =4.2 W =4.5 W =5.0 W =6.0 W =8.0 W =10.0 W =15.0 W C L = 14 L = 16 L = 18 Figure 5.
Top:
Standard deviation of log(
QMI ) as a func-tion of range r for different values of the disorder strength W , for systems of size L = 18. We can see at sufficientlylarge r that the scaling is linear with r . Linear fits at1 (cid:47) W (cid:47) W (cid:39) W . Note that points affected by the finite machineprecision have been removed and that finite size effects arepresent at the largest values of r . Bottom:
Slope C ofthe linear fit of σ [log( QMI )] = C · r + D for different sys-tem sizes. The transition region from C ≈ C ≈ . W our data shows C slowlydropping with W ; it is unclear whether this is affected bythe lack of large r points at large W . At W = 15 we donot have large enough ranges to reach the linear scalingregime. Confidence intervals are defined by the maximum(minimum) value of C found over all linear fits of threeor more consecutive points in the region fitted (see Ap-pendix F). length. For this reason, we anticipate that the decaymust be a stretched exponential out to at least W A min ,which we regard as a lower bound for the value of W at which the decay transitions from stretched expo-nential to exponential: W . Our data suggests thatthat W / < W and W < ∞ in the thermodynamiclimit, a situation where the stretched exponential de-cay region is stable over a region in the MBL phasebefore it becomes an exponential decay. However, wecannot rule out two other scenarios in which either W / → W in the thermodynamic limit or W → ∞ . B. The standard deviation
We use the standard deviation of the distributionsof log(
QM I ) as a measure of their width. It is alreadyapparent from Fig. 2 that the width of the distribu-tions deep in the ergodic phase is constant. Aroundthe transition and in the MBL phase, the width in- W s k e w [ l o g ( Q M I )] L = 183 410 r =1 r =2 r =3 r =4 r =5 r =6 r =7 r =8 r =9 r =10 r =11 r =12 r =13 r =14 r =15 r =16 r =17 W [ l o g ( Q M I )] L = 183 401 r =1 r =2 r =3 r =4 r =5 r =6 r =7 r =8 r =9 r =10 r =11 r =12 r =13 r =14 r =15 r =16 r =17 Figure 6. Third excess moment (skewness; µ ) and fourthexcess moment ( µ ) of log( QMI ) as a function of W forall ranges r for a system of size L = 18. Vertical dashedlines at W = W / (where we independently find β = 1 / W / . Distributions are pos-itively skewed at large W (for large enough r ) and neg-atively skewed at weak W . At the range invariant pointthe skewness is close to -0.65. At the range invariant point µ ≈
0. Higher odd (even )moments show similar behavioras the skewness ( µ ) (see Fig. 7). creases with r .We present σ [log( QM I )] as a function of r in thetop panel of Fig. 5. These curves (after eliminatingdistributions affected by machine precision, and ig-noring finite size effects at large range r ) follow linearscaling as a function of r of the form σ [log( QM I )] = Cr + D . The lower panel of Fig. 5 shows the values of C as a function of W for different system sizes. Con-stant σ [log( QM I )] deep in the ergodic phase gives C = 0. In the MBL phase C (cid:47) .
5, dropping slowly(or staying nearly constant) as W increases. Betweenthese two extremes at small and large W , there is arapid increase in C from C ≈ C ≈ .
5, which getssharper at larger system sizes L . The curves at differ-ent L cross at a value of W which is within the rangeof typically estimated values of the critical W c and wecan treat this as a poor man’s scaling collapse (our at-tempts of carrying out a more formal scaling collapsewere unsuccessful at generating reliable results). C. The skewness and higher moments
Fig. 2 shows that the distributions of log(
QM I ) areskewed negatively deep in the ergodic phase, and pos-itively deep in the MBL phase (at large enough r ),with perhaps a more symmetric, close to unskewed r . L = 14, n = 3 . L = 14, n = 4 . L = 14, n = 5 . L = 14, n = 6 L = 14, n = 713579111315 r . L = 16, n = 3 . L = 16, n = 4 . L = 16, n = 5 . L = 16, n = 6 L = 16, n = 72 3 4 5 W r . L = 18, n = 3 2 3 4 5 W . L = 18, n = 4 2 3 4 5 W . L = 18, n = 5 2 3 4 5 W . L = 18, n = 6 2 3 4 5 WL = 18, n = 7 Figure 7. Colormap of the excess standardized moment µ n for n = 3 , , , , L ; µ n is linearlyinterpolated across W and contour lines are added for clarity; as usual, large W and r distributions that were affected bythe finite machine precision are removed. Vertical, dashed blue lines show the value of W / at which the typical QMI decays as a stretched exponential with β = 1 /
2. Contour lines show µ n becomes range invariant close to W / . Thisagreement is very close at smaller values of n ; at larger values of n , however, we see the agreement becoming closer as L grows. In addition, our results suggest even moments being range invariant at a value of zero (dotted contour line). shape around the transition. In this section we studythe skewness of these distributions, as well as theirhigher-order statistical moments.The excess moment of order n of a distribution overa random variable x is defined as: µ n ≡ E [( x − (cid:104) x (cid:105) ) n ] σ n − µ norm. n = µ stand. n − µ norm. n , (5)where µ stand. n is the standardized moment of order n (normalized by the the n ’th power of the standarddeviation) and µ norm. n is the n ’th moment of the nor-mal distribution, which by definition has all excessmoments equal to zero. µ norm. n is zero for odd n and σ n ( n − n . The third standardized momentis called the skewness .In Fig. 6 we present the skewness ( µ ) and µ oflog( QM I ) as a function of W for each range r for asystem of size L = 18. As expected, the skewnessis negative at small W and positive for large ranges r at large values of W . Interestingly, as seen moreclearly in the inset of Fig. 6, the curves of differentranges cross at a point that is close to W / for L = 18(vertical dashed line), which was estimated indepen-dently in Section II A as the disorder strength at which β = 1 / µ also becomes range invariant close to W / . In addition, the scale invariant value of µ isclose to zero.We now proceed to inspect the excess moments in a more systematic way. Fig. 7 shows colormaps of µ n as a function of W and r . We see that odd (even)moments look alike. In all cases the moments be-come range invariant (at large enough ranges) closeto W / (independently computed in Section II A)with qualitatively different behavior between largerand smaller W . The difference between the apparentrange-invariant value of W and W / decreases quicklywith L ; for odd moments, even at small L , W / is al-ready very close to the range-invariant value of W . In-terestingly, for even moments (but not odd moments),the µ n = 0 contour line is essentially at W / at largeenough r and L = 18. Finally, we note that the rangesat which µ n shows range invariant behavior becomelarger with n ; in addition, in all cases we observe slightfinite size effects at the largest ranges. D. Putting it all together
In summary, the typical correlations in a one-dimensional spin chain of the model in Eq. (1) de-cay exponentially deep in MBL. Deep in the ergodicregion, correlations are constant with range r . Atmoderate disorder strength, and above W c , ( W c ≤ W ≤ W ), typical correlations decay as a stretchedexponential ( QM I typ = e − Ar β ), which takes the form QM I typ = e − A √ r at the transition ( i.e. , W / = W c )and the exponential form ( β = 1) at W . Our resultssuggest both this stretched exponential and exponen-tial decay region of the phase diagram are stable inthe thermodynamic limit [57]The distributions of log( QM I ) have constantspread (standard deviation) deep in the ergodic phase.At moderate and strong disorder strengths, theybroaden linearly with range r .Our results show various similarities between theergodic-MBL transition and the random singlet phase,which emerges as an infinite disorder fixed point instrong disorder renormalization group studies of theground states of disordered spin systems. Typical cor-relations, which decay as a stretched exponential with β = 1 / √ r . Our resultsare consistent with this for all standardized moments(up to the 7’th); note however the standard devia-tion (and also the variance, i.e. , the second moment),does not collapse even under the √ r rescaling. Thismight be regarded as the ergodic-MBL transition sat-isfying a weaker version of universality as conjecturedin Ref. [35] for the random singlet phase. While wefind zero even excess moments, odd moments appearto converge to non-zero values.There is a paradox in the fact that at W / thedistribution of log( QM I ) takes a universal form witha mean that decays with √ r , while its standard de-viation increases as Cr + D . Such family of distri-butions would quickly (as r increases) have half oftheir weight above QM I max = 2 log(2), which is anupper bound for the
QM I . In order for these scal-ings (mean and standard deviation) to be compat-ible with a fixed distribution of log(
QM I ) at longrange, the area under the distribution that lays above
QM I max has to vanish with r , or at least stay con-stant. The only way out of this paradox is a coeffi-cient A ( W = W / , L ) that increases at least as fastas L / with system size, but not with a smaller ex-ponent. This way, larger values of r are only encoun-tered for large values of L , which guarantee a largeenough coefficient A , and thus enough room for thedistribution to broaden while staying mostly belowthe 2 log(2) threshold. Our results (see lower panel ofFig. 4, W / stars) are compatible with this scaling;however, given the small amount of data (only threesmall values of L ), we cannot make any reliable claim.In general, in the stretched exponential decay region,we require A ( W, L ) to scale at least as L − β . III. EXTREME CORRELATIONS
In this section we study the strong tail of the distri-butions of the
QM I , i.e. , the probability that a pairof sites at range r apart has a very large QM I . In con-trast to the typical values of the distribution of Fig. 2that were studied earlier in Section II, we now focuson the upper end of these distributions. Looking at r = | i j |10 p r ( s = . ) L = 18 W =1.0 W =3.0 W =10.0 r = | i j | W = 3.0 L =14 L =16 L =18 Figure 8. Probability p r ( s = 0 .
4) (see Section III) giventwo sites i and j , with r = | i − j | , as a function of r fordifferent disorder strengths W (left) and L (right). Unre-ported data points correspond to values for which we hadno samples and so were unable to estimate p r ( s ). Deep inthe MBL phase ( W = 10), p r ( s ) decays with r , in line withthe localization of correlations. Deep in the ergodic phase( W = 1), correlations are small and have small spreadin their order of magnitude (see Section II) so finding avalue of the QMI that exceeds s is improbable. Aroundthe transition ( W = 3), the probability of finding strongtwo-site QMI bonds becomes range invariant, i.e. , is con-stant as a function of r at sufficiently large r (and awayfrom finite size effects at very large r ). Errors representthe standard deviations of the distribution of p r ( s ) over200 bootstrapping resamples of the disorder realizations.Dotted lines represent the saturated value of p r ( s ), p sat . these extreme values of mutual information is probingrare resonances in the system.To systematically study the strong tail of the distri-butions of the QM I , we define p r ( s ), i.e. , the proba-bility that a pair of sites i and j with range r = | i − j | has a QM I larger than a threshold s . While we aremostly interested in s = log(2) (or s = 2 log(2)) it isimpractical to get statistics on these values of s giventhe rarity of such resonances. Instead, we consider aset of systematically increasing s out to s ≈ . p r ( s ) as a function of r , forfixed s ( s = 0 . p r ( s ) becomesrange invariant around the transition ( W = 3) forlarge enough r (but away from the largest values of r , in order to avoid finite size effects). Deep in theMBL phase, our data shows p r ( s ) decays with r up tothe ranges that we have access to; at more moderatevalues of W , while p r ( s ) gets smaller, our data is notsufficient to distinguish between decaying and range-invariant behavior. In the ergodic phase, p r ( s ) decaysvery rapidly with r leaving us with very few samplesto analyze. That said, the physics of the decay in theergodic phase are fundamentally different than in theMBL phase. In the ergodic phase, p r ( s ) is small due toevery spin being weakly entangled with all other spins,which together with the monogamy of the QM I leavesno room for a strong pairwise mutual information. Inthe MBL phase, p r ( s ) is small because spins at large r are very rarely entangled with each other. This canbe seen from Fig. 2: while the ergodic distributionsof log( QM I ) are compact around a small value of the
QM I the MBL distributions are broad and centered W p s a t ( s , L , W ) L = 18 s =0.05 s =0.10 s =0.15 s =0.20 s =0.25 s =0.30 s =0.35 s =0.40 s W p s a t , m a x l o g ( ) × l o g ( ) W c Mostly many-body resonances Mostly single-particle resonances L = 14 L = 16 L = 18 Figure 9.
Top: p sat ( s, L, W ) (see Sec. III A) as a functionof W . The position of the maxima W max ≡ W p sat,max ( s )(stars) is obtained from a local high-order polynomial fit;we find good results fitting to a polynomial of degree sevenfor all points available in the interval W ∈ [0 , . p sat and W max , represent the standarddeviations of 200 bootstrap resamples over disorder real-izations; the estimation of the W max is highly sensitiveacross resamples for the largest values of s , hence thelarge error bars. Bottom:
Values of W max for different L and their linear extrapolations for large s . Typically,numerical studies find a critical disorder strength of about W c ≈ .
8; as a visual guide, we have shaded in red theregion where W c is thought to be in the thermodynamiclimit. The extrapolations are compatible with W max ≈ W c when s = log(2). around an exponentially small value of the QM I , withvery rare strong values of the
QM I at large range.
A. Proliferation of strong long-rangecorrelations around the transition
To better quantify the behavior of the strong
QM I pairs, we consider the saturation probability p sat ( s, L, W ) as the probability p r ( s ) at large r foreach tuple ( s, L, W ). The value of p sat is shown inFig. 8(right) as a dashed line; the saturation value de-cays slightly with L . We extract p sat ( s, L, W ) fromall values of ( W, L ) where p r ( s ) > r ; thevalue is extracted by averaging p r ( s ) in the interval r ∈ [10 , L − s, L, W ) which are not truly saturated; thiswill not affect the qualitative results we are consid-ering here as we care about the large values of p sat which indeed look convincingly saturated.Fig. 9(top) presents the values of p sat as a functionof W for different thresholds s for a system of size L = 18. We find that the maximum value W max ( s, L ) of p sat ( s, W, L ) (shown by a star) is at a disorderstrength W close to the transition. Notice that the po-sition of W max ( s, L ) becomes unstable at large thresh-old s , due to the small number of samples past thatthreshold as well as the flatness of the curves aroundthe maximum.In the bottom panel of Fig. 9 we plot W max ( s, L )as a function of threshold s for different L , finding W max ( s, L ) rises linearly with s . A particularly in-teresting value of the threshold is s = log 2, whichcorresponds to the value of the pairwise QM I be-tween all pairs of spins in the canonical multi-siteresonating “cat” state: √ ( | Ψ (cid:105) + | Ψ (cid:105) ) where | Ψ (cid:105) and | Ψ (cid:105) are product states which differ in k > s which are this large, a linear extrapolationfinds that W max (log 2) ≈ . L = 18, surprisinglyclose to the best estimates of the transition from scal-ing collapse [6]. This suggests the existence of long-range multi-site resonances at the critical point whoseproliferation has been suggested in being responsiblefor melting MBL; see Ref. [23] for a complementarynumerical approach for probing these long-range res-onaces. IV. EXTREME ENTANGLEMENTEIGENVALUES
In this section, we study λ ≡ − log( ρ ), where ρ isthe second singular value of the reduced density ma-trix of a subsystem over an eigenstate of the Hamilto-nian in Eq. (1). Ref. [56] argues that the probabilityof λ = log 2, i.e. , p ∗ ≡ lim λ → log(2) + p ( λ ) , (6)is finite throughout the MBL phase and zero in theergodic phase, allowing it to be used as an order pa-rameter for the many-body localized phase of matter.Moreover, the authors of Ref. [56] find p ∗ is robust tofinite size effects, showing negligible variations acrossdifferent values of L for small system sizes. Note thatlog(2) is the smallest possible value for λ and corre-sponds to a single singlet entangling the subsystem toits environment. Ref. [56] studies this in the Gaussian-disordered random Heisenberg model, developing evi-dence for this conjecture.In this section, our study differs from Ref. [56] inthree important ways. Instead of the case of randommagnetic fields sampled from a Gaussian distribution,we study the uniform-field case of Eq. (1). Secondly,while Ref. [56] considered subsystems of size 5 (seeAppendix D), we consider subsystems of size L/
2. Fi-nally, to determine p ∗ , Ref. [56] looks at the proba-bility density function (PDF) of λ , in order to de-termine whether p ∗ is finite or zero as it approacheslog(2). In our results, we find this limit of the PDFis very sensitive to the choice of bin size. The finitesize of the bins of our histograms were giving the illu-sion that p ∗ was finite with an estimated value of p ∗ W L = 14 L = 16 L = 18 Figure 10. Exponent γ of Eq. (7). Confidence inter-vals correspond to the standard deviation of γ from 200bootstrap resamples over the original disorder realizations. γ = 1 deep in the MBL phase, which is compatible witha non-zero value of p ∗ (see Eq. (6)) and hence a non-zeroprobability of finding a single singlet across the half cutin the chain. At moderate disorder strengths and close tothe transition γ >
1, which implies a vanishing probabilityof finding a singlet across the chain. Deep in the ergodicphase we do not have enough extremal data to fit a powerlaw close to log(2). See Appendix G for more details onthe extraction of γ and its error bars. which depended on bin-size. To alleviate this problem,we instead consider the cumulative distribution func-tion (CDF), which has no binning and which we findpresents a more robust method to estimate the behav-ior of the distribution in the limit of λ → log(2). Wethen look at the behavior of the CDF measuring theexponent γ of its algebraic approach to log(2), i.e. ,lim λ → log(2) + CDF ( λ ) ∝ [( λ − log(2))] γ . (7)For the PDF of λ to be non-zero as it approacheslog 2 ( i.e. , p ∗ (cid:54) = 0) the CDF has to approach log(2)with γ = 1. On the contrary, γ > p ∗ = 0.Fig. 10 shows the empirical values of γ we find.Deep in the MBL phase (i.e. W > γ ≈ p ∗ > W (although wecan not rule out that even there γ is still marginallyabove 1); therefore, we potentially expect a single sin-glet spanning the two halves of the system deep inthe MBL phase. At moderate values of W but stillin the MBL phase as well as at the transition, γ issignificantly above 1, indicating that p ∗ = 0. LikeRef. [56], we do find that γ shows low sensitivity tosmall changes in system size. This is presumably dueto the fact that singlets across a cut entangle primar-ily, at moderate and strong disorder, spins close tothe cut, which are far away from the boundaries ofthe system and therefore have low sensitivity to itssize (see Appendix A2). V. CONCLUSIONS
Most of this work focuses on the sutdy of the cor-relations throughout the ergodic-MBL phase diagramof the model of Eq. 1 through the distributions of thelogarithm of the
QM I . We have focused on two as-pects of this distribution: the overall shape of the dis-tribution ( i.e its moments) and the extreme tails ofthe distribution. Given the large amount of data (10 eigenstates at energy density (cid:15) ≈ . W, L )) andrelatively large values of L , we can get precise resultson multiple aspects of these distributions.The main contribution of our work is identifyinga region at moderate disorder strength that shows astretched exponential decay ( QM I typ = e − Ar β ) of thetypical correlations (log-averaged QM I ) as a functionof range r . At the transition, this decay takes the form QM I typ = e − A √ r , which is also found in the randomsinglet phase. To further under the universality of thedistribution of log( QM I ) at the transition we considerhigher moments. We find the log(
QM I ) is universalfor all moments except for the second moment, whichscales quadratically with range ( σ [log( QM I )] scaleslinearly). This has some similarities with the distribu-tion of the logarithm of the correlations in the randomsinglet phase which is conjectured to be universal af-ter rescaling by √ r to remove its mean [35] (note thatrescaling by its mean does not affect the scaling ofthe standardized moments). The distributions of thelog( QM I ) seem therefore to satisfy a weaker versionof the universality conjectured for the random singletphase. This aspect of the distributions should be akey feature in determining the universality class of theMBL-ergodic transition and can serve as a constraintfor the panoply of RG results which attempt to explainthe phenomenology of the MBL-ergodic transition.A second key result has involved studying the atyp-ically strong correlations across the system. Our re-sults show the existence of large range-invariant pair-wise
QM I at the transition suggesting the existenceof proliferating multi-site resonances.Finally, we have also studied the extremal values ofthe second entanglement eigenvalue, λ , which signalsthe presence of a single bit of entanglement across abipartition of the system. Our results show that thisprobability is only finite deep in the MBL phase, andvanishes as a power law at moderate values of thedisorder strength and in the ergodic phase. ACKNOWLEDGMENTS
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QMI
The two-site
QM I for a spin- total magnetizationpreserving model like the one in Eq. 1 is computedfrom a few expectation values: (cid:104) n i (cid:105) , (cid:104) n j (cid:105) , (cid:104) σ + i σ − j (cid:105) ,and (cid:104) n i n j (cid:105) , where n i ≡ σ zi +12 . In particular, QM I ij = S i + S j − S ij , (A1)where S i = − (cid:104) n i (cid:105) log( (cid:104) n i (cid:105) ) − (1 − (cid:104) n i (cid:105) ) log(1 − (cid:104) n i (cid:105) )(A2) S ij = − (cid:104) n i n j (cid:105) log (cid:104) n i n j (cid:105)− (cid:104) (1 − n i )(1 − n j ) (cid:105) log [ (cid:104) (1 − n i )(1 − n j ) (cid:105) ] − λ + log( λ + ) − λ − log( λ − ) (A3)with λ ij, ± = (cid:104) ( n i − n j ) (cid:105) ± (cid:113) ( (cid:104) n i (cid:105) − (cid:104) n j (cid:105) ) + 4 |(cid:104) σ + i σ − j (cid:105)| (cid:104) (1 − n i )(1 − n j ) (cid:105) = 1 − (cid:104) n i (cid:105) − (cid:104) n j (cid:105) + (cid:104) n i n j (cid:105) (A5) (cid:104) ( n i − n j ) (cid:105) = (cid:104) n i (cid:105) + (cid:104) n j (cid:105) − (cid:104) n i n j (cid:105) . (A6) Appendix B: Numerical precision in thecomputation of the
QMI
The eigenstates of the model of Eq. 1 are obtainedwith double-precision floating-point numbers, whichmeans that the largest vector entry (assuming it is O (1)) has a precision of ≈ − . This implies the ex-pectation values of Appendix A will also have a preci-sion of about 10 − . When computing the QM I fromthese expectation values, there are two points wherethe precision might drop. First, all terms in Eqs. A2and A3 are of the form x log( x ), with 0 ≤ x ≤ x has a precision of 10 − ) is of order10 − ( e.g. , x log( x ) = − . × − for x = 10 − ),and the QM I overall has a precision of about 10 − .Starting from vectors of precision 10 − (as we do),there is nothing we can do about this drop in preci-sion.There is another point at which precision can drop, i.e. , the computation of λ ± , which involves squaringexpectation values that we have obtained with a pre-cision of 10 − followed by a square root. The squareneeds of a precision of 10 − in order to keep an over-all 10 − after the square root is taken. The useof slightly higher-precision floating-point types suchas numpy.longdouble or numpy.float128 in python does not solve this issue (note these types have typi-cally a precision of 10 − ). We instead make use of the s W W c L = 14 s W c L = 16 s W c L = 18 p s a t ( s , L , W ) / p s a t , m a x ( s , L ) Figure A1. Colormap with the high-order polynomial fitsof the top panel, normalized by the maximum value of p sat . W p sat,max ( s ) is plotted, which is compatible with the max-imum probability of finding strong, long-range QMI bondsat the critical disorder strength W c in the thermodynamiclimit. decimal module in python in order to work with arbi-trary precision; in practice we use 60 decimal places.This gives us confidence that x has precision 10 − in the x log( x ) terms (note we cannot improve thisprecision, given that we start our computation fromdouble-precision vector entries). The QM I ultimatelyhas a precision of ≈ − . Below this threshold, thedistributions of the QM I of Fig. 2 still look smooth,but should not be trusted. In practice we consideronly those distributions which have at least 99% oftheir mass above 10 − , i.e. , an order of magnitudeabove the typical double-precision threshold of 10 − .Finally, we find in practice that 1 − (cid:104) n i (cid:105) and (cid:104) (1 − n i )(1 − n j ) (cid:105) (see Appendix B) are on rare occasionsnegative and of order (cid:47) − . This is expected fromthe precision we work with. Since this is a problemfor the evaluation of their corresponding logarithms,we substitute these values by 10 − . Note that, giventhe magnitude of these terms, this substitution doesnot reduce the precision of the QM I further.
Appendix C: Alternative view of p sat and itsmaxima Fig. A1 shows a colormap with all fitted curves of p sat ( s, W, L ) obtained for thresholds between s = 0and s = 0 . p sat,max , so that they are allvisible in the plot. Appendix D: Second entanglement eigenvalue fora susbsystem of size 5
In Section IV we studied the extremal values of thesecond bipartite entanglement eigenvalue λ over a cutat the middle of the chain. However, Ref. [56] studiedthis quantity over a cut between a subsystem A ofsize L A = 5 and the rest of the chain. In Fig. A2 wepresent the values of γ (see Section IV) for such a cut.There is no substantial difference between this cuts3 W L = 14 L = 16 L = 18 Figure A2. Same as Fig. 10 of the main text for a cutbetween a subsystem A of size L A = 5 and the rest of thechain. Error bars represent the standard deviation of γ across 200 bootstrapping resamples over disorder realiza-tions. and the half-cut of the entanglement entropy. Appendix E: Log-log plots
QMI typ for theextraction of β Figs. A3, A4, and A5 present all linear fits of (cid:104) log(
QM I ) (cid:105) as a function of range r in a log-log scale for all ( L, W ). As discussed in Section II A, this letsus extract the exponent β of the stretch exponentialdecay of QM I typ = e − Ar β . Appendix F: Standard deviations of log(
QMI ) andlinear fits to extract C Figs. A6, A7 and A8 show all linear fits used inSection II B in order to extract the slope C of thelinear scaling of σ [log( QM I )] = Cr + D as a functionof range r . Appendix G: Linear fits to the CDF of the secondbipartite entanglement eigenvalue
Here we present data related to the extraction of theexponent γ for the CDF of the second entanglementeigenvalue, λ , when it approaches λ → log(2) + ,which was discussed in Section IV. In particular,Figs. A9, A10, and A11 linear regression results ofthe fit on a log-log plot to the left-side tails of p ( λ )as a function of λ − log(2). The slope of this fit isequal to γ . In order to estimate the error bars for γ ,we perform a bootstrapping analysis with 200 resam-ples over disorder realizations. The inset of the figuresprovides the distribution of γ from bootstrapping; itsstandard deviation is taken as an estimate of the erroron the estimation of γ .4 l o g ( Q M I t y p ) W = 0.5 = 0.01 W = 1.0 = 0.04 W = 1.5 = 0.17 W = 2.0 = 0.33 l o g ( Q M I t y p ) W = 2.5 = 0.50 W = 2.8 = 0.59 W = 3.0 = 0.64 W = 3.1 = 0.67 l o g ( Q M I t y p ) W = 3.2 = 0.69 W = 3.3 = 0.70 W = 3.4 = 0.72 W = 3.5 = 0.74 l o g ( Q M I t y p ) W = 3.6 = 0.75 W = 3.7 = 0.77 W = 3.8 = 0.78 W = 3.9 = 0.79 l o g ( Q M I t y p ) W = 4.0 = 0.80 W = 4.2 = 0.82 W = 4.5 = 0.85 W = 5.0 = 0.88 r l o g ( Q M I t y p ) W = 6.0 = 0.93 r W = 8.0 = 0.98 r W = 10.0 = 1.01 r W = 15.0 = 1.01 Figure A3. Log-log plots of − log ( QMI typ ) as a function of range r for L = 14. We extract the exponent β of thestretched exponential of Section II from the slope of a linear fit in the interval given by the red vertical lines. We can seethat the fit is good at strong disorder and down to the system size dependent value of W for which β = 0 .
5, correspondingto the ergodic-MBL transition. W / = 2 .
50 for L = 14; we can see that the fit is particularly good around W / , evenat the lowest ranges r . At W < W / the linear fit overestimates log( QMI typ ) at low r , while at W > W / the linearfit underestimates log( QMI typ ) at low r . l o g ( Q M I t y p ) W = 0.5 = 0.00 W = 1.0 = 0.02 W = 1.5 = 0.11 W = 2.0 = 0.27 l o g ( Q M I t y p ) W = 2.5 = 0.43 W = 2.8 = 0.54 W = 3.0 = 0.60 W = 3.1 = 0.63 l o g ( Q M I t y p ) W = 3.2 = 0.65 W = 3.3 = 0.68 W = 3.4 = 0.70 W = 3.5 = 0.71 l o g ( Q M I t y p ) W = 3.6 = 0.73 W = 3.7 = 0.75 W = 3.8 = 0.76 W = 3.9 = 0.77 l o g ( Q M I t y p ) W = 4.0 = 0.79 W = 4.2 = 0.81 W = 4.5 = 0.84 W = 5.0 = 0.87 r l o g ( Q M I t y p ) W = 6.0 = 0.93 r W = 8.0 = 0.98 r W = 10.0 = 1.00 r W = 15.0 = 1.00 Figure A4. Same as Fig. A3 for systems of size L = 16. W / = 2 .
69 for L = 16. l o g ( Q M I t y p ) W = 0.5 = 0.00 W = 1.0 = 0.00 W = 1.5 = 0.06 W = 2.0 = 0.20 l o g ( Q M I t y p ) W = 2.5 = 0.35 W = 2.8 = 0.47 W = 3.0 = 0.54 W = 3.1 = 0.58 l o g ( Q M I t y p ) W = 3.2 = 0.61 W = 3.3 = 0.64 W = 3.4 = 0.66 W = 3.5 = 0.68 l o g ( Q M I t y p ) W = 3.6 = 0.70 W = 3.7 = 0.72 W = 3.8 = 0.73 W = 3.9 = 0.75 l o g ( Q M I t y p ) W = 4.0 = 0.77 W = 4.2 = 0.79 W = 4.5 = 0.82 W = 5.0 = 0.87 r l o g ( Q M I t y p ) W = 6.0 = 0.93 r W = 8.0 = 0.98 r W = 10.0 = 1.00 r W = 15.0 = 0.99 Figure A5. Same as Fig. A3 for systems of size L = 18. W / = 2 .
88 for L = 18. [ l o g ( Q M I )] W = 0.5 W = 1.0 W = 1.5 W = 2.0 [ l o g ( Q M I )] W = 2.5 W = 2.8 W = 3.0 W = 3.1 [ l o g ( Q M I )] W = 3.2 W = 3.3 W = 3.4 W = 3.5 [ l o g ( Q M I )] W = 3.6 W = 3.7 W = 3.8 W = 3.9 [ l o g ( Q M I )] W = 4.0 W = 4.2 W = 4.5 W = 5.0 r [ l o g ( Q M I )] W = 6.0 r W = 8.0 r W = 10.0 r W = 15.0 Figure A6. Linear fits to the standard deviation of log(
QMI ) as a function of r for different values of W and a systemof size L = 14. The linear fits are in general of very good quality, with the exception of 1 (cid:47) W (cid:47) W (cid:39)
8, wherewe only have a few points in the linear scaling regime; at W = 15 we do not have enough points to even enter the linearscaling regime. [ l o g ( Q M I )] W = 0.5 W = 1.0 W = 1.5 W = 2.0 [ l o g ( Q M I )] W = 2.5 W = 2.8 W = 3.0 W = 3.1 [ l o g ( Q M I )] W = 3.2 W = 3.3 W = 3.4 W = 3.5 [ l o g ( Q M I )] W = 3.6 W = 3.7 W = 3.8 W = 3.9 [ l o g ( Q M I )] W = 4.0 W = 4.2 W = 4.5 W = 5.0 r [ l o g ( Q M I )] W = 6.0 r W = 8.0 r W = 10.0 r W = 15.0 Figure A7. Same as Fig. A6 for systems of size L = 16. [ l o g ( Q M I )] W = 0.5 W = 1.0 W = 1.5 W = 2.0 [ l o g ( Q M I )] W = 2.5 W = 2.8 W = 3.0 W = 3.1 [ l o g ( Q M I )] W = 3.2 W = 3.3 W = 3.4 W = 3.5 [ l o g ( Q M I )] W = 3.6 W = 3.7 W = 3.8 W = 3.9 [ l o g ( Q M I )] W = 4.0 W = 4.2 W = 4.5 W = 5.0 r [ l o g ( Q M I )] W = 6.0 r W = 8.0 r W = 10.0 r W = 15.0 Figure A8. Same as Fig. A6 for systems of size L = 18. C D F W = 0.5 W = 1.0 W = 1.5 W = 2.0 p () C D F W = 2.5 p () W = 2.8 p () W = 3.0 p () W = 3.1 p () C D F W = 3.2 p () W = 3.3 p () W = 3.4 p () W = 3.5 p () C D F W = 3.6 p () W = 3.7 p () W = 3.8 p () W = 3.9 p () C D F W = 4.0 p () W = 4.2 p () W = 4.5 p () W = 5.0 p () log(2) C D F W = 6.0 p () log(2) W = 8.0 p () log(2) W = 10.0 p () log(2) W = 15.0 p () Figure A9. Linear fits of the log-log representation of the left-side tail of the CDF of the second eigenvalue of thebipartite reduced density matrix of a system of size L = 14. Red vertical lines denote the ends of the interval thatcontains the points used in each case to make the fit. The slope of the fit gives us the exponent γ in the expression CDF ( λ ) ≈ ( λ − log(2)) γ , which is equivalent to equivalently P DF ( λ ) ≈ ( λ − log(2)) γ − up to an additive constant.Below W = 1 .
5, the low λ data is inexistent and we do not attempt to extract an exponent γ from a linear fit. Inset: probability distribution of γ extracted from 200 bootstrap resamples over the 200K disorder realizations for this systemsize. Bootstrapping was used in order to compute confidence intervals for γ , shown in Fig. 10. C D F W = 0.5 W = 1.0 W = 1.5 W = 2.0 p () C D F W = 2.5 p () W = 2.8 p () W = 3.0 p () W = 3.1 p () C D F W = 3.2 p () W = 3.3 p () W = 3.4 p () W = 3.5 p () C D F W = 3.6 p () W = 3.7 p () W = 3.8 p () W = 3.9 p () C D F W = 4.0 p () W = 4.2 p () W = 4.5 p () W = 5.0 p () log(2) C D F W = 6.0 p () log(2) W = 8.0 p () log(2) W = 10.0 p () log(2) W = 15.0 p () Figure A10. Same as Fig. A9 for systems of size L = 16. Inset: probability distribution of γγ
5, the low λ data is inexistent and we do not attempt to extract an exponent γ from a linear fit. Inset: probability distribution of γ extracted from 200 bootstrap resamples over the 200K disorder realizations for this systemsize. Bootstrapping was used in order to compute confidence intervals for γ , shown in Fig. 10. C D F W = 0.5 W = 1.0 W = 1.5 W = 2.0 p () C D F W = 2.5 p () W = 2.8 p () W = 3.0 p () W = 3.1 p () C D F W = 3.2 p () W = 3.3 p () W = 3.4 p () W = 3.5 p () C D F W = 3.6 p () W = 3.7 p () W = 3.8 p () W = 3.9 p () C D F W = 4.0 p () W = 4.2 p () W = 4.5 p () W = 5.0 p () log(2) C D F W = 6.0 p () log(2) W = 8.0 p () log(2) W = 10.0 p () log(2) W = 15.0 p () Figure A10. Same as Fig. A9 for systems of size L = 16. Inset: probability distribution of γγ extracted from 200bootstrap resamples over the 200K disorder realizations for this system size. C D F W = 0.5 W = 1.0 W = 1.5 W = 2.0 p () C D F W = 2.5 p () W = 2.8 p () W = 3.0 p () W = 3.1 p () C D F W = 3.2 p () W = 3.3 p () W = 3.4 p () W = 3.5 p () C D F W = 3.6 p () W = 3.7 p () W = 3.8 p () W = 3.9 p () C D F W = 4.0 p () W = 4.2 p () W = 4.5 p () W = 5.0 p () log(2) C D F W = 6.0 p () log(2) W = 8.0 p () log(2) W = 10.0 p () log(2) W = 15.0 p () Figure A11. Same as Fig. A9 for systems of size L = 18. Inset: probability distribution of γγ