Probing the randomness of ergodic states: extreme-value statistics in the ergodic and many-body-localized phases
PProbing the randomness of ergodic states: extreme-value statistics in the ergodic andmany-body-localized phases
Rajarshi Pal
Department of Physics, Sungkyunkwan University, Suwon 16419, Korea.
Arul Lakshminarayan
Department of Physics, Indian Institute of Technology Madras, Chennai, India 600036.
The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-bodylocalization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave asrandom states and hence the eigenvalues of the reduced density matrix, commonly referred to as the entan-glement spectrum, are expected to follow the eigenvalue statistics of a trace normalized Wishart ensemble. Inparticular, the density of eigenvalues is supposed to follow the universal Marchenko-Pastur distribution. We finddeviations in the tails both for the disordered XXZ with total S z conserved in the half-filled sector as well as ina model that breaks this conservation. A sensitive measure of deviations is provided by the largest eigenvalue,which in the case of the Wishart ensemble after appropriate shift and scaling follows the universal Tracy-Widomdistribution. We show that for the models considered, in the metallic phase, the largest eigenvalue of the reduceddensity matrix of eigenvector, instead follows the generalized extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution indicating that the correlations between eigenvalues are much weaker compared tothe Wishart ensemble. We show by means of distributions conditional on the high entropy and normalized par-ticipation ratio of eigenstates that the conditional entanglement spectrum still follows generalized extreme valuedistribution. In the deeply localized phase we find heavy tailed distributions and Lévy stable laws in an appro-priately scaled function of the largest and second largest eigenvalues. The scaling is motivated by a recentlydeveloped perturbation theory of weakly coupled chaotic systems. I. INTRODUCTION
The phenomenon of Anderson localisation [1] has beenfound to survive interactions [2, 3] and a flurry of researchactivity has been devoted to understanding and characterisingthis transition to a localized phase from a delocalized or er-godic one. This phenomenon widely referred to as many-bodylocalisation (MBL) is fundamentally interesting as such sys-tems generically break ergodicity and fail to thermalise–thuslying beyond the scope of statistical mechanics. Also, MBLoccurs throughout, and especially the middle of the spectrumimplying that it is an infinite temperature quantum phase tran-sition, different from usual quantum phase transitions studiedat zero temparature[4–7]. These facts combined have signif-icant practical implications for quantum transport [2] and in-formation storage [8, 9]. Experimental advances have allowedthe controlled observation of MBL phenomena [10], furtherdriving interest.Ideas from quantum information have played an importantrole in the development of the understanding of the ergodic-localized transition. Quantum entanglement, a topic of muchimportance in quantum information theory, has also gainedrelevance in quantum many-body physics in the past few years[11, 12]. In particular, the entanglement entropy provides awealth of information about physical states, including novelways to classify states of matter that do not have a local orderparameter [13, 14]. Many-body eigenstates of thermalizingsystems exhibit an entanglement entropy that scales with thevolume of the subregion being considered, while many-bodyeigenstates of many-body localized systems display a bound-ary law scaling (with possible logarithmic corrections). Thestudy of entanglement entropy and its dynamics has played acrucial role in the elucidation of the properties of the local- ized and thermal or ergodic phases. Indeed, studies of en-tanglement entropy [15, 16] provided the first clues as to theemergent integrability of the localized phase. The differencesin the nature of multipartite entanglement in the thermal andlocalized phase is also complementarily reflected in quantitiessuch as a concurrence that measure two qubit entanglements[17]. However, entanglement entropy captures only a smallpart of the full entanglement structure of a system. Muchgreater information is contained in the entanglement spectrum[18], from which the entanglement entropy, and much more,maybe extracted.In the context of MBL, the entanglement spectrum has beenstudied in some recent papers [19–22] and power laws havebeen found in disorder averaged Schmidt eigenvalues plot-ted against the eigenvalue order. In the ergodic phase, dif-ferent statistical properties of energy levels, such as the ra-tio of nearest neighbor spacings ([23]) have been shown tocorrespond to that of one of the canonical random matrix en-sembles, the GOE (Gaussian orthogonal ensemble) [7]. Infact going beyond short range correlations the number vari-ance [24, 25] between levels was computed in [26] and wasshown to be consistent with the corresponding random ma-trix theory (RMT) formulae till about 100 mean-level spac-ings. This together with the claim that the limiting densityof eigenvalues of the reduced density matrices of the eigen-states is identical to the Marchenko-Pastur [19] would indi-cate that deep in the ergodic, delocalized phase, the system iswell described by standard random matrix ensembles. A con-sequence of this is that the eigenstates should be random statesand the reduced density matrix (of say k spins out of a total of L ) belong to the so-called trace-constrained Wishart ensemble MM † / Tr ( MM † ) , where M is a random matrix of dimension2 k × L − k , whose entries are zero-centered independent nor- a r X i v : . [ c ond - m a t . d i s - nn ] F e b mal random numbers. The average entanglement entropy isthe von Neumann entropy of such an ensemble and is givenby the Page value [27].It is known that the largest eigenvalues of the Wishart en-semble after a suitable shift and scaling satisfies the universalTracy-Widom distribution [28], a deviation from the classi-cal extreme value statistics due to the strong correlations ofthe eigenvalues. On the other hand it is also well knownthat if a set of random variables are independent and iden-tically distributed then for appropriately rescaled variablesthere are three possible limiting universal distributions for theextreme maximal events: the Fréchet, Fisher-Tipett-Gumbel,and Weibull distributions [29, 30]. Respectively, they arisedepending on whether the tail of the density is a power law,or faster than any power law, and unbounded or bounded. Ifthere are correlations, then it is known that these universaldistributions are still valid and reached for sufficiently fast de-cay of autocorrelations [31]. Thus that the Tracy-Widom lawdiffers from these distributions is the consequence of the pe-culiar strong correlations present in the eigenvalues of randommatrices.The results of the present paper indicate that the distribu-tion of the maximum of the entanglement spectrum deviatessignificantly from the Tracy-Widom distribution and insteadfits quite well a Fisher-Tipett-Gumbel distribution. This isseen both in a disordered XXZ model with total S z conserved,which has become a standard model for studying the transi-tion and also in a model with an extra field breaking it whichshows the two-component structure [19]. This indicates thatcorrelations in the entanglement spectrum are not Wishart likeeven deep in the delocalized phase, and perhaps shows signsof pre-localization [32]. Most eigenstates in the metallic statehave high entanglement and are delocalized as reflected bythe participation ratio. We look at conditional distributionsof the maximum eigenvalue of the density matrices filteringonly high entropy and participation ratio states and they alsofollow generalized extreme value distribution (GEV) distribu-tions and not the Tracy-Widom statistic. This is seen to holdeven as we start moving towards the transition point but awayfrom it towards the MBL phase, interestingly, the distributionbecomes more Fréchet-like. Even when the maximum distri-bution starts deviating significantly from GEV, we show thatthe conditional distribution still follows GEV, signaling sig-nificant heterogeneity in the spectrum. In the localized phase,we find power laws in the scaled distribution of the largestand second largest eigenvalues. These are quite distinct fromthe power laws in the disorder averaged entanglement spectraitself plotted against the eigenvalue order [21].Previous studies on extreme eigenvalues of the reduceddensity matrices were mostly in the context of strongly chaoticbipartite systems. Remarkable exact results, beyond theaymptotic universal distributions, have been obtained withinrandom matrix theory (RMT) for both the maximum and theminimum eigenvalues and these have been compared withconductance fluctuations in chaotic cavities and entanglementspectra of systems such as coupled kicked tops [33–38]. Astudy of the transition in the distribution of the largest eigen-value of the reduced density matrix of eigenstates ofchaotic bipartite systems as a function of interaction, showed that thelargest eigenvalue showed deviations from statisticality evenwhen the interactions are strong enough for other statistics toshow random matrix behavior [39]. II. PRELIMINARIES: EXTREME-VALUE STATISTICSAND ENTANGLEMENT
This section collects well-known details about extreme-value statistics that are of relevance to this paper, includinga discussion of entanglement in random states and the impli-cations for the extreme eigenvalues of the reduced density ma-trices.
A. Extreme-values of independent or weakly correlatedrandom variables
Let X , X , ..... X N be N observations of an identical inde-pendent random variable x with density p ( x ) . Under very gen-eral conditions we know from the central limit theorem thatthe mean ∑ i X i / N is normally distributed for large N . How-ever, the extremes ( X max ) N = max { X , X , ... X N } and ( X min ) N defined similarly are not normally distributed, but still showuniversal behavior. Let,Prob (( X max ) N < x ) = F N ( x ) = Prob ( X < x , X < x , ... X N < x ) = (cid:18) (cid:90) x p ( x ) dx (cid:19) N . For large N , after a shift and change of scale y = ( x − a N ) / b N , F N ( y ) tends to one of three universal distributions, dependingon the tail of the density p ( x ) [29, 30]. These are(i) Fisher-Tipett-Gumbel: exp ( − exp ( − y )) , if the tail of p ( x ) decays as ∼ e − x δ , δ >
0, faster than a power law,(ii)
Weibull: exp ( − ( − y ) α ) for y ≤ x isbounded above, and the tail of p ( x ) decays as ∼ | x | − − α , α > Fréchet: exp ( − y α ) for y ≥ x isbounded below, and the tail of p ( x ) decays as ∼ x − − α , α > a N and b N depend on the sample size N and α or δ , in particular for the Fisher-Tipett-Gumbel law, a N , which follows on requiring that (cid:82) ∞ a N p ( x ) dx = / N , is ∼ ( log N ) / δ . These distributions can be expressed in a com-bined way through the generalized extreme value distribution(GEV) with the cumulative distribution function, F ( y ; ξ ) = exp ( − ( − ξ y ) / ξ ) for ξ (cid:54) = = exp ( − exp ( − y )) for ξ = , (1)with ξ being the shape parameter, ξ = ξ < ξ > (cid:104) x i x j (cid:105) − (cid:104) x i (cid:105)(cid:104) x j (cid:105) is say exponentiallysmall ∼ e −| i − j | / ξ with ξ (cid:28) N then the extremes still followone of the three classical extreme distributions. The maximumintensity in random states, for example, is an exactly solv-able case of weakly correlated random variables that limit tothe Fisher-Tipett-Gumbel distribution [40]. One well-knowncase where strong correlations lead to a limiting distribution,the Tracy-Widom distribution, different from the above threeclassical ones, are the eigenvalues of random matrices [28].However, for our purposes, we are interested in the singu-lar values of random matrices which are closely connectedto entanglement and the largest values also follow the Tracy-Widom distribution [41]. B. Extreme-value statistics, entanglement and the Wishartensemble
Let us begin by introducing some of the properties of theWishart ensemble of random matrix theory which is the ap-propriate ensemble for modelling statistics of the entangle-ment spectrum. We will discuss the real Wishart ensem-ble throughout, as the Hamiltonians we consider have time-reversal anti-unitary symmetry [20].Let, | Φ (cid:105) be a state belonging to the tensor product space H ⊗ H with dim ( H ) = n , dim ( H ) = n and n ≤ n .In our case, since we will be considering entanglement acrosstwo L / n = n = L / . Let, {| i (cid:105) , | j (cid:105)} be an orthonormal basis in H and H respectively. We have with respect to this basis the state | Φ (cid:105) and its Schmidt decomposition, | Φ (cid:105) = n ∑ i = n ∑ j = c i j | i (cid:105) ⊗ | j (cid:105) = n ∑ i = (cid:112) λ i | i (cid:48) (cid:105) | i (cid:48) (cid:105) . (2)The reduced density matrix of the subsystems are given by ρ = Tr ( | Φ (cid:105)(cid:104) Φ | ) and ρ = Tr ( | Φ (cid:105)(cid:104) Φ | ) . It follows thatSchmidt coefficients λ i are eigenvalues of ρ = CC † , or equiv-alently ρ = ( C † C ) T [42], where C is the “coefficent matrix"with elements c i j .The state is unentangled if and only if λ = λ > λ j = / n for all j . The entanglement en-tropy in the state | Φ (cid:105) is the von Neumann entropy of the re-duced density matrices, S = − Tr ( ρ log ρ ) = − Tr ( ρ log ρ ) .If S =
0, then the state is unentangled, while a maximally en-tangled state has S = log n .Hamiltonians are modeled as random matrices from theGOE in the so called ergodic phase of many-body systems.If the coefficients c i j come from an eigenvector of a typi-cal GOE matrix the induced probability distribution on theSchmidt eigenvalues λ i and the consequences for entangle-ment are well-known. [43, 44] The eigenvectors of a matrix from a GOE of dimension n , are only constrained by normal-ization and the joint distribution of their components is hencegiven by, P ( x , x , ... x n ) = Γ ( n / ) π n / δ (cid:32) ∑ i x i − (cid:33) . (3)Hence, the distribution of c i j is same as that of [ M i j ] / (cid:112) Tr ( MM † ) with M being an unstructured matrix withall elements i.i.d. zero mean normally distributed numbers,the standard orthogonal “Ginibre ensemble” [45]. Thus thereduced density matrices are given by the ensemble of ran-dom matrices, ρ = MM † Tr ( MM † ) . (4)These are the so called trace constrained Wishart ensemble ofrandom matrix theory.The joint probability density function (j.p.d.f.) of λ i , theeigenvalues of ρ , is P ( λ , · · · , λ n ) = B n , n δ (cid:32) n ∑ i = λ i − (cid:33) n ∏ i = λ β ( n − n + ) − i ∏ j < k | λ j − λ k | β , (5)where β = , M and B n , n is a normalization constant known explicitly [24].The symmetry or Dyson index is β = λ i , the delta function is amuch weaker source of correlation and is identical to that ofeigenfunction components as both originate from normaliza-tion [40]. The strong and peculiarly RMT correlations arisefrom the Vandermonde determinant factor involving the prod-uct of the differences of every pair of eigenvalues. In the ab-sence of the j.p.d.f. for the eigenvalues of the reduced densitymatrices of the eigenstates of many-body systems, we want toinvestigate if the consequences of the strong correlations forthe largest eigenvalue are present in the physical systems.However, before delving into this question, we also investi-gate the average entanglement and the density of states (of λ i ),which follow from the j.p.d.f.. While an exact result is knownfor β =
2: (the Page formula) [27], (cid:104) S (cid:105) = n n ∑ k = n + k − n − n ≈ ln n − n n , (6)the asymptotic result for large n and n is valid for both β = β =
1. Marchenko-Pastur Law
The average density of the Schmidt eigenvalues is obtainedby integrating out all variables except one. For n ≥ n (cid:29) Q = n / n ≥
1, the limit of the density ofscaled eigenvalues ˜ λ i = λ i n is given by the Marchenko-Pasturlaw, ρ QMP ( x ) = Q π (cid:112) ( x + − x )( x − x − ) x , x − ≤ x ≤ x + , and 0 otherwise. The distribution is in the finite support [ x − , x + ] , where x ± = + / Q ± / √ Q .For the case Q =
1, especially relevant for the numericalresults presented, the distribution is given by, ρ MP ( x ) = π (cid:114) − xx , ≤ x ≤ , (7)and zero otherwise. The distribution thus diverges at the ori-gin. The Marchenko-Pastur law is in fact a universal distribu-tion for ensembles of correlation matrices, irrespective of theexact distribution of matrix elements as long as it has a finitemoments of sufficiently larger order [46]. The moments ofthe Marchenko-Pastur distribution, M n are given by the Cata-lan numbers, (cid:104) x n (cid:105) = C n = n + (cid:18) nn (cid:19) , thus M / nn → n → ∞ [47].
2. Distribution of maximum eigenvalue
As mentioned earlier, the distribution of the maximumeigenvalue of a Wishart ensemble after suitable centering andscaling follows the Tracy-Widom distribution [28] which hasno simple closed-form. For the orthogonal case, it can bedefined implicitly by the Hastings-McLeod solution to thesecond Painlevé equation, [28]. For our purposes, since thereduced density matrix corresponding to a random state hasunit trace, we need to adapt the results of the Wishart ensem-ble to the trace constrained one. This was done in [48], forcomplex Wishart matrices. Adapting the methods used therewith the results of real Wishart matrices in [41] we obtain thesame center and scaling in the large n limit as complex matri-ces, namely a centering or shift of 4 / n and a scaling equal to2 n − . This is obtained as follows.We have our reduced density matrix ρ = W / S (with W = M † M and S = Tr ( W ) , Eq. (4)) and thus λ max ( ρ ) = λ max ( W ) / S .Now, from [41] we know that mean of the distribution of λ max ( W ) , (cid:104) λ max ( W ) (cid:105) goes as ( √ n − + √ n ) while mean ofthe distribution of trace of W goes as (cid:104) S (cid:105) ∼ n . Hence, we ap-proximately have the mean of λ max ( ρ ) equal to, (cid:104) λ max ( ρ ) (cid:105) ∼ ( √ n − + √ n ) n = n , for large n . Again from [41] we know thatthe standard deviation of the distribution of λ max ( W ) , σ λ max ( W ) ∼ ( √ n − + √ n )( √ n − + √ n ) . As, the fluctuation in the largest eigenvalue is much greaterthan the fluctuation in the sum of eigenvalues, i.e, σ λ max ( W ) (cid:104) λ max ( W ) (cid:105) (cid:29) σ S (cid:104) S (cid:105) we approximately have, σ λ max ( ρ ) = σ λ max ( W ) (cid:104) S (cid:105) = n − for large n . III. NUMERICAL RESULTS OF THE STATISTICS INTWO SPIN MODELS
Model:
We consider the following model for MBL, whichis an XXZ spin- chain of L spins with a random z field and asmall constant x field. H = L − ∑ i = (cid:0) σ xi σ xi + + σ yi σ yi + + ∆ σ zi σ zi + (cid:1) + ∑ i h i σ zi + Γ ∑ i σ xi (8) h i chosen to be i.i.d. uniformly random in [ − W , W ] . We con-sider both Γ = Γ = .
1. For the Γ = S z is conserved and we restrict ourselves to the half-filled sectorwith open boundary conditions. Since, we are interested instudying how random the model truly is in the ergodic phase,we need to break the S z conservation without affecting thetransition too much to see the effect of this conservation onthe randomness.We will refer to the model with Γ = Γ = . H and H . The model H has been extensively stud-ied and is believed to capture all essential properties of theMBL phase and the localization transition. For example, itis known that the model supports a MBL phase at strong dis-order, an ergodic phase at weaker disorder, and an integrablepoint at zero disorder. For ∆ = W ≈ . H was found.Throughout the paper we have considered the middle one-third eigenstates for data. We have checked that there is nosignificant difference in the distributions we have computedif we instead choose a single eigenstate from the middle ofthe spectrum and many more disorder realizations. 500 disor-der realizations are chosen for the L = H model and 100disorder realizations for L = H model.Let X be a discrete random variables with outcomes1 , , ... n and let p i = P ( X = i ) . The Rényi entropy of order α , where α ≥ α (cid:54) = H α ( X ) = − α log (cid:32) n ∑ i = p α i (cid:33) . In the limit of α → ∞ the Renyi entropy converges to the min-entropy, H ∞ ( X ) = min i {− log p i } = − log max i p i . (9) Scaled eigenvalue − − − − − L og ( d e n s i t y ) L = 14 , H L = 12 , H L = 14 , H MP L = 14 , Normalized Wishart − − FIG. 1. Log of the scaled eigenvalue distribution for different L for H and H for L = W = .
5, MP refers to the Marchenko-Pasturdistribution in Eqn. (7).The tail is shown in the inset.
The min-entropy, as its name indicates, is the smallest of theRényi entropies and is the most conservative estimate of theinformation content of a random variable. In this paper westudy the maximum of the entanglement spectrum, which es-sentially provides us with the min-entropy of entanglement.
A. Deviations from the Marchenko-Pastur distribution
We first compare the average density of the entanglementspectrum with the Marchenko-Pastur (MP) distribution. Thisis computed for W = . L for H and H in Fig. (1). In the figures, scaled eigenvalue refers to theeigenvalues multiplied by the dimension 2 L / . For L =
14, inthe H model 20 disorder realizations have been used.As is clear, breaking the total S z conservation brings the dis-tributions considerably closer to the MP distribution. Whilethe density approaches the MP distribution for larger valuesof L , the tails show that the limiting distribution is perhapsclose to MP, but different. Similar observations were reportedin [22]. The Marchenko-Pastur has finite support from 0 to 4,but the distributions obtained from the data show exponentialtails. Of course, the hard bound is obtained in the asymptotic L → ∞ limit, however the softening of this for finite L [35] istoo small in comparison to the tails observed here.This is also reflected, as it should be, in the deviations ofthe moments from that of the MP distribution. The deviationsfrom this are shown in Fig. (2) for H , with the blue andorange curves representing respectively M k / k of the distribu-tion obtained for L = , W = .
1. Deviation from gaussianity and normalized participation ratio
While looking for deviations shown by the entanglementspectrum statistics from the predictions of Wishart ensemble, k )051015202530 M k / k M k /k , H C k /k FIG. 2. Plot of M k / k with log ( k ) for L = , H and W = . C k / k with log ( k ) andsaturates to 4. − − − − √ nx . . . . . . D e n s i t y L = 12 , H L = 14 , H L = 12 , H Std. normal5 10 − FIG. 3. Distribution of scaled eigenvector components √ nx for L = , H and L = , H and disorder strength W = .
5. The insetshows the tail, the y-axis of the inset represents the logarithm of thedensity. it is natural to ask if the normalized components of eigenvec-tors ( √ nx i , see Eqn. (3)) are themselves normally distributed,to begin with. While this would be the case if the GOE en-semble were to apply, although this is not a necessary condi-tion for the MP distribution, due to its universality. As seenin Fig. (3), while the distributions approach Gaussian for in-creasing L , the fact that the logarithm of the distribution hasnear linear rather than quadratic tails, as seen in the inset, indi-cates that the limiting distribution is perhaps different. Inter-estingly, as seen in Fig. (4) the distribution of the normalizedcoefficients of the eigenvectors fits an exponential power dis-tribution with p.d.f. P ( x ) = β α Γ ( / β ) exp (cid:18) − (cid:12)(cid:12)(cid:12) x α (cid:12)(cid:12)(cid:12) β (cid:19) (10) − − − − √ nx . . . . . . D e n s i t y Gennorm fitL=12,W=0.5
FIG. 4. Distribution of scaled eigenvector components √ nx for L = , H and W = .
5, compared to a generalized normal distributionas in Eq. (10). quite well. This is a generalization of the normal distribution( β =
2) with an additional shape parameter β . The β for thefits in the figure is to the leading order 1 . α = . n ∑ i ( √ nx i ) , which is just the normalized inverse partici-pation ratio , n ∑ i x i with respect to the product basis in which H and H are diagonalized. Here we compute the inverse ofthis quantity, the normalized participation ratio (PR) which isthe inverse of the kurtosis for different states. This mean valueof the PR is known to be equal to 1 / ≈ ln n − [27]. In Fig. (5)we show the entropy vs. PR plots for H and H for L = H and H are respectively,0 .
259 and 0 . . .
5, while the Page value ≈ .
66. Also as is clear fromFig. (5), in the model with total S z conservation the PR andentropy have lower variance and are closer to random valuesas expected. B. Deviations of the maximum from the Tracy-Widomdistribution
Following the adaptations mentioned before, here we com-pare the maximum eigenvalue data after using a center andscaling respectively of n and 2 n − with n = L / . We have, λ (cid:48) = ( λ − n ) n − . (11) FIG. 5. Entanglement entropy vs participation ratio for L = , H ,500 realizations and L = , H ,100 realizations and W = . − Maximum eigenvalue . . . . D e n s i t y Tracy-WidomL=12Normalized Wishart
FIG. 6. Comparison of the distribution of the shifted-and-scaledmaximum eigenvalue, λ (cid:48) , Eq. (11), with the Tracy-Widom distri-bution for L = , H and W = .
5. Shown for comparison is alsothe case of a (trace normalized) Wishart ensemble of dimension2 L / = In order to compare for finite size effects, we also plot datafrom a trace normalized Wishart ensemble of the same dimen-sions. This is shown in Fig. (6). The Tracy-Widom distribu-tion is obtained by using the R package called RMTstat [52].As is clear, there are considerable deviations much be-yond the finite size effects from the Tracy-Widom distribution,which implies that the correlations between eigenvalues of thereduced density matrix are not Wishart like. Thus, surpris-ingly even though the NNS distribution and number varianceof the levels match GOE predictions for H and H ([7, 26]), amore rigorous test with respect to extreme statistics shows thatthe correlations between eigenvalues of the reduced densitymatrix of eigenvectors of the spin chains are much weaker. InFig. (7) we show a fit of the data with the generalized extremevalue distribution with the probability density function, f ( y ; ξ ) = exp ( − ( − ξ y ) / ξ )( − ξ y ) ( / ξ − ) for ξ (cid:54) = = exp ( − exp ( − y )) exp ( − y ) for ξ = . . (12) − λ max . . . . . P ( λ m a x ) GEV fit L = 12 , H , W = 0 .
50 50 . . FIG. 7. Fit with the generalized extreme value distribution of thedensity of λ (cid:48) max obtained from Hamiltonian H ( L =
12) and Hamil-tonian H ( L = W = . The shape parameter ξ takes a value of 0 .
069 and − . L = H data and L = H data for W = .
5. The data has been centered and scaled by the lo-cation and scale parameters obtained by fitting a generalizedextreme value distribution with free location and scale param-eters (using Scipy) so that Eq. (12) can be used. In all theplots involving GEV we plot λ (cid:48) max = ( λ − loc ) / scale withthe location (loc) and scale parameters for the fit producedin Table I. Note, that as the location and scale parametersfor fitting the Tracy-Widom distribution, given by Eq. (11)are different from that of the GEV distribution, λ (cid:48) max and λ (cid:48) are in general different. For the L = , H data the locationand scale parameters obtained from the fit are respectively0 . , .
007 while for the L = , H data they are respec-tively 0 . , . H model, for L =
12 with entropy and PR respectively greaterthan 3 . .
25. We then try to fit the maximum of the en-tanglement spectrum obtained from this data to the GEV dis-tribution. The result is shown in Fig. (8), and the ξ value forthe fit is 0 .
021 (location and scale parameters for the fit re-spectively being 0 . , . FIG. 8. Fit with the generalized extreme value distribution of thedensity of λ (cid:48) max obtained from Hamiltonian H for W = . L = . .
25. Insetshows the entropy vs. PR scatter plot.
IV. EXTREME VALUE STATISTICS AWAY FROM THEERGODIC PHASEA. Persistence of GEV for moderate disorder strengths
Figure 9 inset shows the entropy-PR scatter plot for the H model for disorder strength of W = .
0. The entropy and PRdistribution spreads considerably with the mean entropy andmean PR respectively lowering to become 3 .
37 and 0 .
17 re-spectively. The variance of entanglement entropy peaks as oneapproaches the transition, and thus the distributions are broad-ened. However, interestingly as shown in the main part of thesame figure, the maximum still fits a GEV well, but with the ξ parameter being equal to − . W , the FIG. 9. Fit with the generalized extreme value distribution of thedensity of λ (cid:48) max obtained from Hamiltonian H for W = . L = vs PR for L =
12, the Hamiltonian H , W = . W = . − λ max . . . . P ( λ m a x ) GEV fit L = 12 , H FIG. 10. Fit with the generalized extreme value distribution of thedensity of λ (cid:48) max obtained from Hamiltonian H for W = . . . λ (cid:48) max obtained from H for W = . L =
12. Entropyvs PR scatter plot shown in inset for L =
12, the Hamiltonian H , W = . W = . eigenstates include many more low entropy and PR statesindicating that they are almost localized and the maximumdistribution seems to stop fitting a GEV. This is shown inFig. (11) for W = .
0. However, if we again weed outstates with an entropy and PR respective lower than 3 . .
1, the distribution again moves close to a GEV with ξ = − . ξ , and locationand scale parameters used for centering and scaling the databefore using Eq. (12) for the different fits for L = H modelare collected in table I. Beyond about W = . − λ max . . . . . P ( λ m a x ) GEV fit L = 12 , H FIG. 12. Fit with the generalized extreme value distribution of thedensity of λ (cid:48) max obtained from H for W = . L =
12 for eigenstateswith entropy greater than 3 . . ξ value for GEV fit W Filtered ξ loc scale . − .
086 0 .
069 0 . . − .
021 0 .
073 0 . . − . .
092 0 . . − .
129 0 .
090 0 . . − .
004 0 .
103 0 . B. Distribution of the maximum and second maximum of theentanglement spectrum and power laws in the MBL phase
For the sake of compactness and simplicity we present re-sults only for the model H without total spin or particle num-ber conservation, although we have verified the same for the H case as well. Figures (13) and (14) show the distribution ofunscaled or shifted (“raw") largest and second largest eigen-value of the reduced density matrices, λ and λ , for W = . H model. While for W = . λ is rather broad, compared to theergodic cases for W = . λ =
1, in-dicating the extreme nature of the extreme, as the other eigen-values are then forced to be of far lesser significance. Thedominance of the largest eigenvalue is indicative of entry intoMBL regimes. A feature, we mention in passing, in the distri-bution of the maximum is the kink that develops at λ = / W = .
0. Such a feature has also been seen in dis-order averaged entanglement entropy as a resonance at ln 2in [53, 54], as well in weakly coupled chaotic systems [39],originating in fact from the behavior of the dominating largesteigenvalue.As the disorder is increased the entanglement tends to thearea-law and the largest eigenvalue tends to 1. As for theergodic phase, with increasing disorder, the GEV statisticsseems to apply well only if we filter states that are sufficientlyergodic, we expect that in the transition regime and in theMBL phase itself it will be harder to control this and as thelargest eigenvalue has become O ( ) rather than of O ( / n ) , we . . . . . λ P ( λ ) W=1.5W=2.0W=2.5W=3.00 . . . . . λ P ( λ ) W=3.5W=4.0W=4.5W=6.0
FIG. 13. Distribution of the largest eigenvalue λ for the Hamiltonian H , with L =
12 as the disorder strength is increased across the MBLtransition. . . . . . . λ P ( λ ) W=0.5W=1.0W=1.5W=2.0W=2.5 .
00 0 .
01 0 .
02 0 .
03 0 .
04 0 .
05 0 . λ P ( λ ) W=3.0W=3.5W=4.0W=4.5W=6.0
FIG. 14. Distribution of λ for H , L = ( ˜ λ ) − − − − P ( ˜ λ ) FIG. 15. Distribution of ˜ λ for the Hamiltonian H , and L =
12, withthe disorder strength W =
6, the slope is ≈ − . ( ˜ λ ) − − − − P ( ˜ λ ) FIG. 16. Distribution of ˜ λ for the Hamiltonian H , and L =
12, withthe disorder strength W =
6, the slope is ≈ − . shift our analysis away from the GEV framework. For largedisorder strengths W , we may take the view that the system isone consisting of spins with a Hamiltonian ∑ i h (cid:48) i σ zi with h (cid:48) i be-ing uniformly distributed in [ − , ] being subjected to weakinteractions of the order 1 / W due to the other terms. Thusthe unperturbed Hamiltonian with W = ∞ has a Poisson spec-trum on which there is weak coupling that leads to resonancesbetween the bare states.Recently a theory for such a scenario has been developed in[39] albeit in bipartite systems of weakly interacting chaoticsystems. Notably, the noninteracting case there leads to aPoisson spectrum (despite the chaos), and the largest and sec-ond largest eigenvalues of the reduced density matrices ofeigenstates were heavy tailed, including the Lévy distribu-tion. In the present case of many-body MBL, the partition isbipartite as well, while the individual subsystems in the non-interacting case are not chaotic, but show Poisson statistics.The key elements of the analysis rely more on the Poisson na-ture of the uncoupled systems and hence it is interesting tocompare the extreme value results from there. We will ignorean overall scaling by a “transition parameter" that is yet to beidentified, if at all it exists, in the case of the MBL transition.Stable Lévy laws were seen, as a consequence of combininga regularized perturbation theory with the generalized centrallimit theorem [39] in the densities of somewhat transformed0quantities ˜ λ = g ( λ ) and ˜ λ = g ( λ ) where g ( x ) = x ( − x )( − x ) . (13)Notice that g ( x ) = g ( − x ) and that for λ , as it is typicallyclose to 1 this is essentially 1 − λ , and for λ which is typi-cally (cid:28)
1, this is ≈ λ itself. However, we are really interestedin the excursions of these values into non-typical values whichhappens frequently due to the power-laws. Note also that λ ≤ /
2. Thus, applying the same transformation, Fig. (15)shows distribution for ˜ λ for the H model for W = .
0, fairlydeep in the MBL regime. An excellent power-law ∼ x − . seems to be obtained, while Figure (16) shows the distribu-tion of ˜ λ , for the H model and once again a power law tailis evident and is close to ∼ x − . .These are close indeed to theones derived in [39] which is ∼ x − / corresponding to boththe cases. Thus the extreme value statistics of MBL are alsodescribed by stable Lévy laws that manifest clearly on an ap-propriate transformation. There is no doubt that the compar-isons with the perturbation results derived in [39] need to bemore critical, but we are encouraged by the unmistakable sim-ilarities with the distribution of say λ as it undergoes a tran-sition into a localized regime, the broadening and the peak at1 / W . V. CONCLUSIONS
In this work we have probed the randomness of the eigen-states of disordered spin chains showing the many-body local-ization transition, in the ergodic phase. We have used extremestatistics of the entanglement spectrum for this purpose. Thisis much more sensitive to the randomness of eigenstates com-pared to the distribution of the average density of the entangle-ment spectra which follows nearly a Marchenko-Pastur distri-bution for random states. For random states, the eigenvaluescome from a trace constrained Wishart ensemble and the max-imum follows the Tracy-Widom distribution after a suitableshift of center and scaling. We have found significant devia-tions from the Tracy-Widom law and the distribution we ob-tained instead fits the Fisher-Tippet-Gumbel distribution quitewell. This is the distribution that maximal events of indepen-dent and identically distributed random variables follow aftersuitable rescaling. Our results thus indicate that even deep inthe metallic phase the correlations in the entanglement spec- tra are not as strong as those coming from truly random statesand perhaps show signs of pre-localization. A natural ques-tion is if this is due to the presence of low entropy and par-ticipation ratio eigenstates present among the states which arecloser to being random. But we have found that even the con-ditional distributions obtained after weeding out such statesfollow the generalized extreme value distribution and not theTracy-Widom statistic. Even when the maximum distributionstarts deviating significantly from GEV as we move towardsthe transition point, we found that the conditional distributionstill follows GEV.The fact that the Tracy-Widom law is not obtained even inthe ergodic phase of a model with no conservation laws otherthan energy, maybe a generic feature of many-body systems.While quantities like the nearest neighbor spacing distribu-tions or ratio of spacings may still be that of random matrices,the deviations will show in such properties of the eigenstatesand the extreme value statistics could be the most stringent testof randomness, or at least one of them. It is possible thoughthat further breaking conservation laws by Floquet kicked sys-tems or even non-periodically driven systems may restore theextreme values to be of the Tracy-Widom type. Excellentagreement with the Marchenko-Pastur law and the Page valueof subsystem entropy have been noted in these cases, for ex-ample see [55].In the localized phase we shifted attention, motivated byrecent results from a perturbation analysis of weakly cou-pled chaotic systems. Remarkably, we found that suitablytransforming the largest and second largest eigenvalues leadto power-law distributions that match reasonably well withthat derived earlier and indicate the applicability of Lévy sta-ble laws in this context. This may also indicate the applica-bility of a regularized perturbation theory in the deep MBLregime, if not close to the transition. A work has appearedsince the beginning of ours with identical motivations [56].While our results unambiguously indicate the presence of theFisher-Tippett-Gumbel law in the ergodic phase, the corre-sponding work shifts attention to the transition or localizedregime. We have instead focused on an alternative strategyin the deep MBL phase wherein there are power-laws in thedistribution of extreme eigenvalues (quite distinct from powerlaws in disorder averaged entanglement spectra itself plottedagainst the eigenvalue order as found in [21]). However, whilewe found it untenable with our data to fit GEV distributionsfor large values of W , the results of [56] indicate that this maystill be possible. Apart from a closer comparison with suchworks, ours would hopefully contribute to an understandingof extreme value statistics in the spectra of many-body sys-tems. [1] P. W. Anderson, “Absence of diffusion in certain random lat-tices,” Phys. Rev. , 1492–1505 (1958).[2] D.M. Basko, I.L. Aleiner, and B.L. Altshuler, “Metalâ ˘A ¸Sinsu-lator transition in a weakly interacting many-electron systemwith localized single-particle states,” Annals of Physics ,1126 – 1205 (2006). [3] John Z. Imbrie, “Diagonalization and many-body localizationfor a disordered quantum spin chain,” Phys. Rev. Lett. ,027201 (2016).[4] Rahul Nandkishore and David A. Huse, “Many-body localiza-tion and thermalization in quantum statistical mechanics,” An-nual Review of Condensed Matter Physics , 15–38 (2015), https://doi.org/10.1146/annurev-conmatphys-031214-014726.[5] David A. Huse, Rahul Nandkishore, and Vadim Oganesyan,“Phenomenology of fully many-body-localized systems,” Phys.Rev. B , 174202 (2014).[6] Jens H. Bardarson, Frank Pollmann, and Joel E. Moore, “Un-bounded growth of entanglement in models of many-body lo-calization,” Phys. Rev. Lett. , 017202 (2012).[7] Arijeet Pal and David A. Huse, “Many-body localization phasetransition,” Phys. Rev. B , 174411 (2010).[8] Laumann C.R Yao N.Y and Vishwanath A., “Many-body lo-calization protected quantum state transfer,” arXiv:1508.06995(2015).[9] M. Serbyn, M. Knap, S. Gopalakrishnan, Z. Papi´c, N. Y. Yao,C. R. Laumann, D. A. Abanin, M. D. Lukin, and E. A. Demler,“Interferometric probes of many-body localization,” Phys. Rev.Lett. , 147204 (2014).[10] Michael Schreiber, Sean S. Hodgman, Pranjal Bordia, Hen-rik P. Lüschen, Mark H. Fischer, Ronen Vosk, Ehud Alt-man, Ulrich Schneider, and Immanuel Bloch, “Observa-tion of many-body localization of interacting fermions in aquasirandom optical lattice,” Science , 842–845 (2015),http://science.sciencemag.org/content/349/6250/842.full.pdf.[11] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Ve-dral, “Entanglement in many-body systems,” Rev. Mod. Phys. , 517–576 (2008).[12] J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Arealaws for the entanglement entropy,” Rev. Mod. Phys. , 277–306 (2010).[13] Alexei Kitaev and John Preskill, “Topological entanglement en-tropy,” Phys. Rev. Lett. , 110404 (2006).[14] Xiao-Gang Wen, “Quantum orders in an exact soluble model,”Phys. Rev. Lett. , 016803 (2003).[15] Jens H. Bardarson, Frank Pollmann, and Joel E. Moore, “Un-bounded growth of entanglement in models of many-body lo-calization,” Phys. Rev. Lett. , 017202 (2012).[16] Jonas A. Kjäll, Jens H. Bardarson, and Frank Pollmann,“Many-body localization in a disordered quantum ising chain,”Phys. Rev. Lett. , 107204 (2014).[17] Soumya Bera and Arul Lakshminarayan, “Local entanglementstructure across a many-body localization transition,” Phys.Rev. B , 134204 (2016).[18] Hui Li and F. D. M. Haldane, “Entanglement spectrum as a gen-eralization of entanglement entropy: Identification of topolog-ical order in non-abelian fractional quantum hall effect states,”Phys. Rev. Lett. , 010504 (2008).[19] Zhi-Cheng Yang, Claudio Chamon, Alioscia Hamma, and Ed-uardo R. Mucciolo, “Two-component structure in the entangle-ment spectrum of highly excited states,” Phys. Rev. Lett. ,267206 (2015).[20] Scott D. Geraedts, Rahul Nandkishore, and Nicolas Regnault,“Many-body localization and thermalization: Insights from theentanglement spectrum,” Phys. Rev. B , 174202 (2016).[21] Maksym Serbyn, Alexios A. Michailidis, Dmitry A. Abanin,and Z. Papi´c, “Power-law entanglement spectrum in many-bodylocalized phases,” Phys. Rev. Lett. , 160601 (2016).[22] Francesca Pietracaprina, Giorgio Parisi, Angelo Mariano, Save-rio Pascazio, and Antonello Scardicchio, “Entanglement criti-cal length at the many-body localization transition,” Journal ofStatistical Mechanics: Theory and Experiment , 113102(2017).[23] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, “Distribu-tion of the ratio of consecutive level spacings in random matrixensembles,” Phys. Rev. Lett. , 084101 (2013).[24] Madan Lal Mehta, Random Matrices , 3rd ed. (2004). [25] Marie-Joya Giannoni, Andre Voros, and Jean Zinn-Justin,
Chaos and quantum physics,Volume 52 of Les Houches Sum-mer School Proceedings Series (North-Holland, 1991).[26] Corentin L. Bertrand and Antonio M. García-García, “Anoma-lous thouless energy and critical statistics on the metallic sideof the many-body localization transition,” Phys. Rev. B ,144201 (2016).[27] Don N. Page, “Average entropy of a subsystem,” Phys. Rev.Lett. , 1291–1294 (1993).[28] Craig A. Tracy and Harold Widom, “On orthogonal and sym-plectic matrix ensembles,” Communications in MathematicalPhysics , 727–754 (1996).[29] R. A. Fisher and L. H. C. Tippett, “Limiting forms of the fre-quency distribution of the largest or smallest member of a sam-ple,” Mathematical Proceedings of the Cambridge Philosophi-cal Society , 180 (1928).[30] E J Gumbel, Statistics of extremes (Dover PublicationsInc.,New York, 2004).[31] M. R. Leadbetter and Holger Rootzen, “Extremal theory forstochastic processes,” The Annals of Probability , 431–478(1988).[32] A. De Luca and A. Scardicchio, “Ergodicity breaking in amodel showing many-body localization,” EPL (EurophysicsLetters) , 37003 (2013).[33] Satya N. Majumdar, Oriol Bohigas, and Arul Lakshminarayan,“Exact minimum eigenvalue distribution of an entangled ran-dom pure state,” Journal of Statistical Physics , 33–49(2008).[34] Satya N. Majumdar, Extreme eigenvalues of Wishart matrices:application to entangled bipartite system,
Oxford Handbook ofRandom Matrix Theory (Oxford University Press, 2011).[35] Hiroto Kubotani, Satoshi Adachi, and Mikito Toda, “Exact for-mula of the distribution of schmidt eigenvalues for dynamicalformation of entanglement in quantum chaos,” Phys. Rev. Lett. , 240501 (2008).[36] Pierpaolo Vivo, “Largest schmidt eigenvalue of random purestates and conductance distribution in chaotic cavities,” Journalof Statistical Mechanics: Theory and Experiment , P01022(2011).[37] Santosh Kumar, Bharath Sambasivam, and Shashank Anand,“Smallest eigenvalue density for regular or fixed-trace complexwishart–laguerre ensemble and entanglement in coupled kickedtops,” Journal of Physics A: Mathematical and Theoretical ,345201 (2017).[38] Peter J Forrester and Santosh Kumar, “Recursion scheme for thelargest $\beta$ -wishart–laguerre eigenvalue and landauer con-ductance in quantum transport,” Journal of Physics A: Mathe-matical and Theoretical , 42LT02 (2019).[39] Steven Tomsovic, Arul Lakshminarayan, Shashi C. L. Srivas-tava, and Arnd Bäcker, “Eigenstate entanglement betweenquantum chaotic subsystems: Universal transitions and powerlaws in the entanglement spectrum,” Phys. Rev. E , 032209(2018).[40] Arul Lakshminarayan, Steven Tomsovic, Oriol Bohigas, andSatya N. Majumdar, “Extreme statistics of complex random andquantum chaotic states,” Phys. Rev. Lett. , 044103 (2008).[41] Iain M. Johnstone, “On the distribution of the largest eigenvaluein principal components analysis,” The Annals of Statistics ,295–327 (2001).[42] Michael A. Nielsen and Isaac L. Chuang, Quantum Computa-tion and Quantum Information: 10th Anniversary Edition , 10thed. (Cambridge University Press, New York, NY, USA, 2011).[43] Karol Zyczkowski and Hans-JÃijrgen Sommers, “Induced mea-sures in the space of mixed quantum states,” Journal of Physics A: Mathematical and General , 7111–7125 (2001).[44] I. Bengtsson and K. ˙Zyczkowski, Geometry of Quantum States:An Introduction to Quantum Entanglement (Cambridge Univer-sity Press, 2017).[45] Jean Ginibre, “Statistical ensembles of complex, quaternion,and real matrices,” J. Mathematical Phys. , 440–449 (1965).[46] Terence Tao and Van Vu, “Random covariance matrices: Uni-versality of local statistics of eigenvalues,” Ann. Probab. ,1285–1315 (2012).[47] F. Haake, Quantum Signatures of Chaos , Physics and astron-omy online library (Springer, 2001).[48] Ion Nechita, “Asymptotics of random density matrices,” An-nales Henri Poincaré , 1521–1538 (2007).[49] David J. Luitz, Nicolas Laflorencie, and Fabien Alet, “Many-body localization edge in the random-field heisenberg chain,”Phys. Rev. B , 081103 (2015).[50] Nicolas Regnault and Rahul Nandkishore, “Floquet thermaliza-tion: Symmetries and random matrix ensembles,” Phys. Rev. B , 104203 (2016).[51] T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey,and S. S. M. Wong, “Random-matrix physics: spectrum andstrength fluctuations,” Rev. Mod. Phys. , 385–479 (1981).[52] Iain M. Johnstone, Zongming Ma, Patrick O. Perry, andMorteza Shahram, RMTstat: Distributions, Statistics and Testsderived from Random Matrix Theory (2014), r package version0.3.[53] David J. Luitz, “Long tail distributions near the many-body lo-calization transition,” Phys. Rev. B , 134201 (2016).[54] S. P. Lim and D. N. Sheng, “Many-body localization and tran-sition by density matrix renormalization group and exact diag-onalization studies,” Phys. Rev. B , 045111 (2016).[55] Sunil K. Mishra and Arul Lakshminarayan, “Resonance andgeneration of random states in a quenched ising model,” EPL(Europhysics Letters) , 10002 (2014).[56] Wouter Buijsman, Vladimir Gritsev, and Vadim Cheianov,“Gumbel statistics for entanglement spectra of many-body lo-calized eigenstates,” Phys. Rev. B100