Characterization of many-body mobility edges with random matrices
CCharacterization of many-body mobility edges with random matrices
Xingbo Wei,
1, 2, 3, 4
Rubem Mondaini, and Gao Xianlong Zhejiang University, Hangzhou 310027, China Beijing Computational Science Research Center, Beijing 100193, China Westlake University, Hangzhou 310024, China Department of Physics, Zhejiang Normal University, Jinhua 321004, China
Whether the many-body mobility edges can exist in a one-dimensional interacting quantum systemis a controversial problem, mainly hampered by the limited system sizes amenable to numericalsimulations. We investigate the transition from chaos to localization by constructing a combinedrandom matrix, which has two extremes, one of Gaussian orthogonal ensemble and the other ofPoisson statistics, drawn from different distributions. We find that by fixing a scaling parameter,the mobility edges can exist while increasing the matrix dimension D → ∞ , depending on thedistribution of matrix elements of the diagonal uncorrelated matrix. By applying those results toa specific one-dimensional isolated quantum system of random diagonal elements, we confirm theexistence of a many-body mobility edge, connecting it with results on the onset of level repulsionextracted from ensembles of mixed random matrices. Introduction.—
A single-particle mobility edge char-acterizing the separation in energy between extended andlocalized states occurs in a variety of non-interactingquantum models, as in one-dimensional ones with long-range hoppings [1, 2], certain incommensurate modula-tions of the potential [3–7], or in three-dimensional dis-ordered systems at a finite disorder strength [8, 9]. Thefate of mobility edges in the presence of interactions is ofgreat interest to experimentalists and theorists, but as oftoday, a large debate questions the existence of its many-body analogue. A variety of numerical results show themanifestation of many-body mobility edges in finite sizes,and some studies employing scaling analysis also arguethat they should occur when approaching the thermody-namic limit [10–20]. Roeck et al. , for example, contendthis view, asserting that localized states will be eventu-ally thermalized by ergodic ones, ruling out their con-comitant appearance and consequently the occurrence ofmany-body mobility edges [21, 22]. Thus, due to the sizelimitations used in all numerical calculations, consensushas not been achieved.To provide a new angle in this controversy, we willstep back, and rather than following the standard proce-dure of analyzing a typical physical model, we will headtowards the theory of random matrices, which providesthe basis of our understanding of quantum chaotic be-havior and its eventual absence [23, 24]. These are nat-urally characterized by the rigidity of the matrix spec-trum, with the former displaying a characteristic levelrepulsion that is not present in regular systems, which inturn, exhibit completely uncorrelated eigenvalues. Whendefining a gap between adjacent eigenvalues in the spec-trum S j ≡ E j +1 − E j , chaotic and non-chaotic behaviorscan be characterized by probability densities of the gaps P GOE ( S ) = ( πS/
2) exp ( − S π/
4) and P P ( S ) = e − S , re-spectively, for unity mean spacing, when dealing withsymmetric matrices [25]. Often what is done in studyingthe many-body localization (MBL) transition [26–29] for physical systems is to combine both types of matrices,gradually changing their relative weight until the statis-tics of the spectrum suddenly changes [30]. If increasingthe weight of the matrix displaying Poisson statistics, thissignals the onset of non-ergodic behavior once the eigen-values of the combined matrix become uncorrelated, andtheir eigenfunctions display support not scaling with thematrix dimension [24]. When purely dealing with ran-dom matrices, this type of combined ensemble mixingmatrices of different symmetry classes has been studiedin the past [31–36], with corresponding surmises for thelevel spacing distributions being specifically defined interms of the weights, and interpolating between P GOE ( S )and P P ( S ), if, e.g., adding random matrices from theGaussian orthogonal ensemble (GOE) and diagonal ran-dom matrices.In this Letter, we employ the mixed ensemble model tostudy the tuning process from GOE to Poisson statistics,unlike others with intermediate ensemble models [37–39],providing strong evidence on the existence of the mobil-ity edges and the stability of the coexistence of localizedand delocalized states in approaching the thermodynamiclimit. Model.—
We construct a combined random matrix M , which reads M = (1 − k ) M e + k M i , (1)where M e is a GOE random matrix constructed by meansof a normal random distribution matrix A by ( A + A T ) / M i is a diagonal matrix with random elements, and k ∈ [0 ,
1] is a real coefficient. The limiting values k = 0( M = M e ) and 1 ( M = M i ) characterize chaotic and non-chaotic regimes, where the probability distributions ofgaps in the spectrum follow P GOE and P P , respectively.Our main interest is focused on the intermediate case0 < k <
1, where M can potentially describe the transi-tion from chaos to localization. In this regime of mixedensembles, surmises for the probability distribution have a r X i v : . [ c ond - m a t . d i s - nn ] J a n . . . . . . α ( a )1 2 3 4 ξ . . . . . . h I i ( d ) ( b )1 2 3 4 ξ ( e ) D = 500 D = 1000 D = 2000 ( c )1 2 3 4 ξ ( f )0 . . . . . . Figure 1. Phase diagrams and corresponding average (overthe whole spectrum) inverse participation ratio (cid:104)I(cid:105) as a func-tion of ξ for different M i s. The diagonal elements of M i are drawn from a Gaussian distribution in (a) and (d); uni-form distribution in (b) and (e); cosine function F ( x ) = √ πx ) with a random number x ∈ [0 ,
1] in (c) and (f).Color bar represents the value of IPR. Disorder realizationshave been performed. The ordinate α = i/ ( D −
1) is the nor-malized level position in (a), (b) and (c) with i = 0 , , ..., D − α = 0 .
01. Reddashed lines mark the positions of ξ ( ξ = 0 . ξ = 3 in (c) and (f)), further analyzed in Fig. 2. been derived [32, 33, 35], in direct similarity with theWigner surmises of pure ensembles [23]. In what follows,we investigate two main indicators of chaotic matrices,as to classify both the eigenvalues spectrum as well astheir eigenfunctions. For the first, we notice that thegap distributions albeit precise in characterizing the er-godic properties of the spectrum, suffer from technicaldifficulties related to the necessity of having mean levelspacing (cid:104) S (cid:105) constrained to unity for the aforementioneddistributions to be valid. Thus instead, we look at thedistribution P ( r ) of the ratio of two consecutive gaps inthe spectrum r α = min { S α , S α +1 } / max { S α , S α +1 } , re-quiring no unfolding schemes [40].In turn, a direct measure of the (de)localization of theeigenfunctions is obtained by the inverse participationratio (IPR), I ( α ) = (cid:80) n | ψ ( α ) n | , where ψ ( α ) n is the α -theigenstate of the matrix and n is the basis state index.The eigenstate properties are characterized according toits limiting values aslim D →∞ I ( α ) ∝ (cid:40) /D, chaotic (extended)const ., localized (2)where D is the matrix dimension. The IPR of a chaotic(extended) eigenstate is D -dependent, in contrast to alocalized one; therefore, a finite-size scaling is necessary − . − . − . − . . l og ( I ) ( a ) D = 125 D = 250 D = 500 D = 1000 D = 2000 . . . . . α . . . . . h r i ( c ) ( b )0 . . . . . α ( d ) Figure 2. Inverse participation ratio I ( α ) and correspondingaverage ratio of adjacent energy gaps (cid:104) r (cid:105) as a function of α for different D s. ξ = 0 . M i of Gaussiandistribution; ξ = 3 for (b) and (d) with with M i from thecosine distribution function. The purple dashed line marks (cid:104) r (cid:105) = 0 .
53 (GOE) whereas the orange dashed one marks (cid:104) r (cid:105) =2 ln 2 − to differentiate the two types of states when approach-ing the limit D → ∞ . In that regime, it is importantto extract a scaling parameter that can account for themodifications the mixed ensemble suffers. For example,the spectral variance of M e is approximately proportionalto D for large enough matrix dimension [36]. For that,we conjecture the following scaling parameter [41], ξ = k ˜ σ (1 − k ) Dσ , (3)where ˜ σ and σ are the standard deviation of the diago-nal elements in M i and M e , respectively. When k changesfrom 0 to 1, ξ changes from 0 to infinity. Without lossof generality, we set ˜ σ = σ = 1 and choose the stan-dard deviation of non-diagonal elements √ / M e . We find that, by changing k and D , but fixing ξ , the system keeps the same localand chaotic properties.We test this scaling form by employing specific distri-butions for the diagonal elements in M i , Gaussian, uni-form and cosine distributions in Fig. 1 [(a),(d)], 1 [(b),(e)]and 1 [(c),(f)], respectively. As shown in Fig. 1(d),1(e), 1(f), the average inverse participation ratios overthe whole spectrum, (cid:104)I(cid:105) , are completely coincident for D = 500, 1000 and 2000 when using the scaling param-eter ξ . This is indicative that for large matrices, the I ( α ) diagram for different distributions Fig. 1(a), 1(b),1(c) (for D = 2000) correctly reflect the phase diagramsfor D → ∞ (See Fig. 2 for details of IPR before theaverage for the different D s). Interestingly, the chaoticregion shows a “D”-like shape in Fig. 1(a), and a “C”-like shape in Fig. 1(c). Since out of the closure of “C”and “D” it is a localized phase, both situations indicate a . . . . . . α ( a )2 4 6 8 10 ξ . . . . . . h I i ( c ) ( b )4 8 12 16 20 ξ ( d ) L = 12 L = 14 L = 16 L = 18 . . . . . . Figure 3. The I ( α ) phase diagrams for L = 18 and (cid:104)I(cid:105) asa function of ξ for different L s at half filling. (a) and (c)for G of Gaussian distribution; (b) and (d) for G of cosinefunction distribution, both with ˜ σ = 1 .
0. Color bar representsthe value of I . The ordinate window ∆ α = 0 .
01 in (a) and(b). Red and black dashed lines mark the positions of ξ ( ξ =2 . , . ξ = 4 . , . typical phase diagram of mobility edges. However, thereis no obvious mobility edges for the uniformly distributed M i [Fig. 1(b)]. Thus, those are model-dependent and doexist for M i of Gaussian distribution or of cosine distri-bution for fixed ξ even when the dimension of the matrixgoes to infinity. Moreover, the transition from classicalchaos to localization with increasing ξ has characteristicsof a crossover.We further study the IPR and (cid:104) r (cid:105) with fixed ξ but dif-ferent D s in Fig. 2. The latter, apart from finite size fluc-tuations, is coincident for increasing dimensions D . Formatrices M i from the Gaussian distribution, the states atthe edges of the spectrum ( α (cid:46) .
05 and α (cid:38) .
95, with ξ = 0 .
1) do not follow GOE statistics, indicating thatthese are not chaotic. This is supported by the corre-sponding values of the IPR with increasing matrix sizesin Fig. 2(a): Only at the central part of the spectrumscaling with D occurs. Conversely, the situation is op-posite for M i constructed from a cosine function distri-bution [See Figs. 2(b) and 2(d)]. At the edges of thespectrum, I ( α ) decreases with increasing dimensions andis accompanied by a mean ratio of adjacent gaps charac-teristic of chaotic matrics from the GOE ensemble for thevalue of ξ = 3 used. Based on these results for mixed en-sembles of random matrices, we apply similar formalismto Hamiltonian matrices of typical quantum systems. Many-body mobility edges.—
In order to test ouranalysis and to determine whether a many-body mobilityedge can exist in a one-dimensional quantum system on a lattice, we construct a many-body model, composed of N spinless fermions in a chain with L sites. The Hamil-tonian is given by ˆ H = ˆ T + ˆ V , (4)where ˆ T is the (non-random) nearest-neighbor hoppingmatrix, ˆ T = − t L (cid:88) i =1 (cid:16) ˆ c † i ˆ c i +1 + H . c . (cid:17) , (5)assuming ˆ c L +1 = ˆ c . The spectral variance of ˆ T is pro-portional to N for large matrix sizes D = (cid:0) LN (cid:1) and fixedfilling n = N/L , as σ E = 2 N (1 − n ), which contrastswith a typical GOE matrix where σ E ∝ D . In turn, thesecond matrix is diagonal, and characterizes both the in-teraction and disorder, which is typically the case forFock bases in real space. It assumes a form ˆ V = V G ,with V its strength, and G populated with random num-bers selected from a particular distribution. Motivatedby the cases where mobility edges were found when deal-ing with pure random matrices, we focus on two types:Gaussian and cosine distributions, both characterized bya standard deviation ˜ σ of its elements. Other distribu-tions, as power-law, can also result in mobility edges andare analyzed in the Supplemental Material [42]. We no-tice that short-range homogeneous interactions result indiagonal matrices whose distribution of elements is givenby a discrete Gaussian, and in the case the interactionsare fully random, they approach a continuous distribu-tion that can also lead to the onset of MBL [43, 44]. Fur-thermore, a typical Anderson-like disorder with uniformrandom local energies also result in a Gaussian distribu-tion of entries in the diagonal elements for a many-bodybasis.The scaling parameter of Eq. (3) in this model be-comes, ξ = V ˜ σtN , (6)which we use, as before, to scale the average values of IPRfor increasing matrix sizes. Hereafter, the filling numberis set to n = N/L = 0 .
5, which results in the largestpossible matrix sizes for a given L .In Fig. 3, we calculate the phase diagrams of the IPRfor L = 18 and its average value, (cid:104)I(cid:105) , as a function of ξ for different L s. The phase diagrams in Fig. 3(a), 3(b)are similar to those of Fig. 1(a), 1(c) for the pure randommatrices, showing robust many-body mobility edges forGaussian and cosine function distributions of diagonalmatrix elements, respectively. In Fig. 3(c), (cid:104)I(cid:105) showsconverged values for different system (matrix) sizes for ξ (cid:38) L here is no longer an important factor to trigger MBL. For ξ < (cid:104)I(cid:105) decreases as L increases, a signal that ergodic . . . . . I ( α ) ( a ) L = 12 L = 14 L = 16 L = 18 . . . . . I ( α ) ( c )0 . . . . . α . . h r i ( e ) ( b )( d )0 . . . . . α ( f ) Figure 4. IPR and (cid:104) r (cid:105) as a function of α for different L s.(a) and (c) correspond to Fig. 3(c) with ξ = 7 . ξ = 2 . ξ = 16 . ξ = 4 .
0, respectively. (e) and (f) are the corresponding (cid:104) r (cid:105) value of (c) and (d), respectively. The red dashed linesmark IPR separation points and the black dashed lines mark (cid:104) r (cid:105) separation points in (c), (d), (e), and (f). In (c) and (e),the red dashed lines are at α = 0 .
06 and α = 0 .
94, and theblack dashed lines α = 0 .
15 and α = 0 .
85. In (d) and (f),the red dashed lines are at α = 0 .
40 and α = 0 .
60 and blackdotted lines α = 0 .
20 and α = 0 . states start to populate the spectrum. Figure 3 (d) forthe cosine distribution shows very similar behavior, butwith a critical ξ (cid:39)
16 separating the regime with thepresence of ergodicity and full MBL.In Fig. 4, we investigate the system size effects on I ( α ) and (cid:104) r (cid:105) across the spectrum α , for different typical valuesof ξ in both distributions. For the many-body localizedregion [typical value ξ = 7 (16) for Gaussian (cosine) dis-tributions], Fig. 4 (a) and (b) show that completely coin-cident IPR for different sizes confirm localization acrossthe whole spectrum. For smaller weights ξ of the diagonalmatrices, ergodic and delocalized states naturally occurin the central part (edges) of the spectrum for G of theGaussian (cosine function) distribution. These regionsare identified by an L -dependence in both I ( α ) and (cid:104) r (cid:105) ,with the former decreasing with the system size and thelatter heading towards (cid:104) r (cid:105) GOE for increasing L s. Whenthese quantities display system size independence, lackof ergodicity and localization ensues. We notice, how-ever, that the onset of this behavior is not coincident in α , i.e., it suggests that the many-body mobility edge oc-curs not as a sharp transition in the spectrum but ratheras one characterized by an intermediate phase with non-ergodic but delocalized wavefunctions. This regime hasbeen found in physical models and dubbed non-ergodicmetal [12, 45–47]. Mobility edges and predictions of mixed ensembles.—
Mixed ensembles provide further fundamental insightson the onset of ergodicity. Starting from a Poisson diag-onal matrix, any non-zero weight of an added GOE ma-trix triggers a finite degree of level repulsion [32, 35], adefining characteristic of ergodicity. However, the chaoticeffects induced by the latter crucially depend on the den-sity of states. That is, the effective weight of the GOEmatrix on the resulting mixed matrix grows as denserthe spectrum is [35], initially investigated via the analy-sis of the density-of-states-resolved statistics of the gaps P ( ρ, S ), which requires unfolding of the spectrum. An r -statistic analysis avoids this inconvenience, but a sur-mise for P ( r ) in the mixed GOE-Poisson ensemble is yetelusive [36]. Nonetheless, we can infer similar informa-tion regarding the effect of the GOE random matrix by aworking numerical definition. Starting from random 3 × M = M i + λ M e , onecan obtain the numerical surmise P ( r, λ ) by using a setof 2,000 of such matrices for each λ . Subsequently, weextract the adjacent gap distribution for a much largercombined D × D matrix with a typical weight λ = Λ. Bydividing the spectrum in a large number of homogeneousenergy windows, we resolve the distribution by the den-sity of states ρ , obtaining P ( r, ρ ). Finally, the effect ofthe weight λ of the chaotic matrix can be quantified as, χ ≡ (cid:88) r [ P × ( r, λ ) − P × ( r, ρ )] , (7)which is shown in Fig. 5, as a color plot. The minimaof this work definition (white markers) assures that theeffect of the λ weight of the GOE matrix is linearly pro-portional to the density of states, a result also valid forthe distributions of the gaps P ( ρ, S ) [35]. We now ap-ply the same analysis in the case of a physical system[Eq. (4)]. Now, the different density of states results in aremarkable contrasting behavior for the effect of the ther-malizing matrix. λ approaches zero at finite density ofstates, which in practice results in the absence of ergod-icity at the edges of the spectrum, that are typically lessdense. This thus indicates that a sharp transition occursin the ergodic properties of the eigenvalues in the spec-trum, where a region with small density of states does notsuffer from thermalizing effects and can be associated tothe manifestation of a mobility edge. Summary and discussion.—
We study the processfrom GOE to Poisson statistics by constructing a com-bined random matrix as to shed light on the long last-ing debate on the existence of mobility edges. We findphase diagrams in the thermodynamic, determined bythe invariant scaling parameter ξ , suggesting its existencefor different distributions of the diagonal random matrix.Lastly, by employing a numerical analysis on the effect ofthe strength of the GOE matrix in different parts of thespectrum, we infer that less dense regions have an effec-tive vanishing weight of the ergodic matrix, albeit denser .
00 0 .
15 0 . ρ ( E )0 . . . λ ( a ) 0 .
00 0 .
01 0 . ρ ( E )( b ) . . . . . . Figure 5. The distance χ between the numerical surmise of3 × M i + Λ M e (Λ = 1 /D ), whereas in (b) for the physicalsystem ˆ H = ˆ T + ˆ V , with ξ = 2 and L = 18 – different systemsizes are attempted in [42]. ones do render ergodic behavior. Thus, we argue thatmobility edges can exist in physical systems with short-ranged hoppings, even when approaching the thermody-namic limit. A closed form of the probability distributionof the ratio of adjacent gaps of mixed ensembles may re-sult in an even more systematic way to define whetherdifferent quantum systems may or not display mobilityedges and will be reserved for a future work.The authors acknowledges insightful discussions withT. Wettig, W. Zhu, and M. Gong. GX and WX acknowl-edge support from NSFC under Grants No. 11835011 andNo. 11774316. RM acknowledges support from NSFCGrants No. U1930402, No. 11674021, 11851110757 andNo. 11974039. 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Supplementary Materials:Characterization of many-body mobility edges with random matrices . . . . . . α ( a )2 4 6 8 10 ξ . . . . . . h I i ( b ) L = 12 L = 14 L = 16 L = 18 . . . . . . I ( α ) ( c ) L = 20 . . α . . h r i ( d )0 . . . . . . Figure S1. The phase diagram for L = 18 (a) and average IPR (cid:104)I(cid:105) (b) as a function of ξ for different L ’s with G populatedwith elements from a power-law function distribution. Thered dashed lines mark ξ = 9 .
0, used in the remaining panels.Spectrum resolved IPR and (cid:104) r (cid:105) for ξ = 9 . (cid:104) r (cid:105) = 0 .
53, and orange onethe correspondent Poisson value (cid:104) r (cid:105) = 2 ln 2 − Many-body mobility edges for other distribution of thediagonal random matrix: Power-law function.—
In themain text, we have analyzed how the distribution of therandom diagonal elements affect the overall shape of thelocalization-delocalization transition in the spectrum-disorder amplitude ( α vs. ξ ) space. In both cases wherea mobility edge was observed, Gaussian and cosine func-tion distribution, it is manifested mainly at the edges ofthe spectrum, where the density of states is small. Finitesize effects are thus more dramatic at those regimes, as in-creasing L ’s can significantly alter the sparsity of the lev-els at the ends of the spectrum. Since the ergodic-MBLtransition shape significantly depends on the random dis-tribution used to construct G , we can proceed with theexercise of finding another distribution whose manifesta-tion of a mobility edge avoids this problem. A naturalchoice, beyond the ones we have chosen, is a power-lawfunction, which we select as F ( x ) = ( x − / / (4 / x is a random number, picked in the interval [ − , /
5, and 4 /
15 are the mean value and standard de-viation of F ( x ) = x , respectively.In Fig. S1(a), we report the phase diagram for thephysical model (4), with L = 18, and elements in G selected according to the probability distribution F ( x )above. There is a clear many-body mobility edge with only one ergodic-to-MBL transition, and the criticalpoint is around the middle of the spectrum for ξ = 9.Fig. S1(b) indicates that the average IPR, (cid:104)I(cid:105) , possessesa small L -dependence in a large range of ξ , in similarityto the other distributions studied in Fig. 3. However,the spectrum resolved IPR, I ( α ) , displays, on the otherhand, a much more sharp transition in the spectrum ap-proaching zero much faster than for other distributionsstudied in the main text. This sharp transition on thelocalization properties of the eigenfunctions is accompa-nied by a well marked transition on the level repulsion[Fig. S1(d)], displaying a GOE-to-Poisson transition forthe statistics of the level spacings at similar values of α . Scaling parameter ξ and relation to previous results.— In Ref. 36, Chavda et al. used a mixed ensemble of theform H λ = H + λ V , (8)where H is the Poisson matrix and V the correspondingGOE one. Similarly to what we did, they selected thevariance of the off-diagonal elements of the GOE matrixbeing one-half of the variance of the diagonal ones, σ .After that, by arriving on a non-closed form of the jointprobability of the eigenvalues of a 3 × λ σ d , (9)where d is the mean level spacing between nearest eigen-values of the diagonal (unperturbed) Poisson matrix. Tomatch our definition of the mixed ensemble, one can eas-ily identify that λ = − kk and that the mean level spacingof the Poisson matrix d = ˜ σ √ D . With this, the scalingparameter we used and theirs are related as, ξ = 1 √ Λ . (10)As us, they identify the very small dependence of theaverage ratio of adjacent gaps over the whole spectrumon the matrix size D , when scaling with the transitionparameter Λ, and we do so by looking at the averagevalue of the IPR, scaling it with ξ (see Fig. 1). The effective ergodic weight in different parts of thespectrum.—
Figure 5 in the main text shows that for amixed ensemble, the effective weight of the ergodic ma-trix on the eigenvalues of the mixed ensemble cruciallydepends on the density of the states of the latter. Thatis, for a denser region of the spectrum, the ergodicity setsin more strongly and is finite almost as any finite den-sity is obtained for the eigenvalue spectrum. For the case .
00 0 . ρ ( E )0 . . . λ ( a ) D = 500 D = 1000 D = 2000 D = 4000 .
00 0 . ρ ( E )0 . . . . . . λ ( b ) L = 12 L = 14 L = 16 L = 18 Figure S2. The effective weight of the ergodic matrix in thecombined ensemble as a function of the normalized densityof states: The goal is to analyze the effects of matrix finite-ness. In (a), for the combined matrix M i + Λ M e (Λ = 1 /D ),whereas in (b) for the physical system ˆ H = ˆ T + ˆ V , with ξ = 2.As in the main text, the markers are obtained by the mini-mization of the distance χ between the numerical surmise insmall matrices and the one obtained for the different matrixsizes, resolving by the density of states. The lines in panel (a)are linear fittings when neglecting the vanishingly small den-sities. In (b), large fluctuations can also occur in this regime,in special for the small matrix sizes. − . . . E . . . . . ρ ( E ) D = 2000 −
50 0 50 E . . . . . ρ ( E ) L = 18 Figure S3. The normalized density of states of the combinedensemble M i + Λ M e (Λ = 1 /D = 1 / H = ˆ T + ˆ V , with ξ = 2 and L = 18 (rightpanel). The critical density of states marking the regime be-low which the effective weight λ of the ergodic matrix becomesnegligible is marked by the horizontal dashed lines. of the physical system, where the dense GOE-matrix issubstituted by a sparse non-random (but symmetric ma-trix), there is a critical density above which the ergodicaspects become relevant. We now analyze the matrixsize dependence on these result in Fig. S2. For the caseof the pure mixed ensemble, Fig. S2(a) shows that thelinear relationship between the effective weight and the(normalized) density of states is valid for a wide rangeof matrix sizes D . Similarly, when using the physicalmodel ˆ H = ˆ T + ˆ V [Fig. S2(b)], the finite size effects donot qualitatively alter the main aspect expressed in themain text, in which a critical level density is necessaryfor the effective ergodic weight to become relevant. It isworth mentioning that the critical density is similar inall system sizes analyzed.Finally, to see that this threshold density of eigenlevelsis not vanishingly small, we report in Fig. S3 the nor-malized density of states of both cases, marking with ahorizontal dashed line the maximum density where theeffective weight λλ