Multifractality meets entanglement: relation for non-ergodic extended states
MMultifractality meets entanglement: relation for non-ergodic extended states
Giuseppe De Tomasi
1, 2 and Ivan M. Khaymovich T.C.M. Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom Department of Physics, Technische Universit¨at M¨unchen, 85747 Garching, Germany Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187-Dresden, Germany
In this work we establish a relation between entanglement entropy and fractal dimension D ofgeneric many-body wave functions, by generalizing the result of Don N. Page [Phys. Rev. Lett. 71,1291] to the case of sparse random pure states (S-RPS). These S-RPS living in a Hilbert space ofsize N are defined as normalized vectors with only N D (0 ≤ D ≤
1) random non-zero elements.For D = 1 these states used by Page represent ergodic states at infinite temperature. However, for0 < D < N D /N of the full Hilbert space. Both analytically and numerically, we show that the mean entanglemententropy S ( A ) of a subsystem A , with Hilbert space dimension N A , scales as S ( A ) ∼ D ln N forsmall fractal dimensions D , N D < N A . Remarkably, S ( A ) saturates at its thermal (Page) value atinfinite temperature, S ( A ) ∼ ln N A at larger D . Consequently, we provide an example when theentanglement entropy takes an ergodic value even though the wave function is highly non-ergodic.Finally, we generalize our results to Renyi entropies S q ( A ) with q > D = 1. Introduction – The success of classical statisticalphysics is based on the concept of ergodicity, which allowsthe description of complex systems by the knowledge ofonly few thermodynamic parameters [1, 2]. In quantumrealm the paradigm of ergodicity is much less understoodand its characterization is now an active research front.The most accredited theory, which gives an attempt toexplain equilibration in closed quantum systems, relieson the eigenstate thermalization hypothesis (ETH) [3–6]. ETH assets that the system thermalizes locally at thelevel of single eigenstates and has been tested numericallyin a wide variety of generic interacting systems [6, 7].It is now well established that entanglement playsa fundamental role on the thermalization process [7–11]. Thermal states are locally highly entangled withthe rest of the system, which acts as a bath. Conse-quently, the measurement of entanglement entropy (EE)has been found to be a resounding resource to probeergodic/thermal phases, both theoretically [12–17] andrecently also experimentally [18–21]. For instance, infi-nite temperature ergodic states are believed to behavelike random vectors [3, 7] and their EE reaches a precisevalue often referred as Page value [22].On the other hand, ergodicity is deeply connected tothe notion of chaos [7, 23], which implies also an equipar-tition of the many-body wave function over the availablemany-body Fock states, usually quantified by multifrac-tal analysis, e.g., by scaling of the inverse participationratio (IPR) [24]. In this case, infinite temperature ergodicstates span homogeneously the entire Hilbert space [25].The latter states should be distinguished from the so-called non-ergodic extended (NEE) states. These NEEstates live on a fractal in the Fock space, which is avanishing portion of the total Hilbert space. Recently,the NEE have been invoked to understand new phasesof matter like bad metals [26–34], which are neither in-sulators nor conventional diffusive metals and also are found in chaotic many-body quantum system like in theSachdev-Ye-Kitaev model [35–37].Very recently, the two aforementioned probes, EE andIPR, have been used to describe thermal phases (speciallyat infinite temperature), and to detect ergodic-breakingquantum phase transitions [12, 38–41]. Nevertheless, therelations between these two probes has not been stud-ied extensively so far [42]. Thus, the natural questionarises: to what extend do these probes lead to the samedescription?In this work, we build up a bridge between ergodicproperties extracted from EE and the ones from multi-fractal analysis. With this aim, we generalize the semi-nal work of Page [22], computing EE and its fluctuationsfor NEE states. Remarkably, we show, both analyticallyand numerically, that a subsystem EE can still be er-godic (Page value), even though the states are highlynon-ergodic. Consequently, the mean value of EE mightbe not enough to state ergodicity, though EE reaches thePage value.
General definitions – The Renyi entropy, S q ( A ), of asubsystem A with Hilbert space dimensions N A = N p , p ≤ /
2, is defined as: S q ( A ) = ln Σ q − q , with Σ q = Tr A [ ρ qA ] = N A (cid:88) M =1 λ qM , (1)where ρ A = Tr B [ ρ ] is the reduced density matrix ofthe subsystem A , after tracing out the subsystem B = A c and { λ M } are Schmidt eigenvalues of ρ A . Thevon Neumann EE, S ( A ) = lim q → S ( A ) equals to − Tr A [ ρ A ln ρ A ]. For a pure state ρ = | ψ (cid:105) (cid:104) ψ | , ρ AM,M (cid:48) = N B (cid:88) m =1 ψ M,m ψ ∗ M (cid:48) ,m , (2)where ψ M,m are the wave function coefficients | ψ (cid:105) = (cid:80) N A − M =0 (cid:80) N B − m =0 ψ M,m | M (cid:105) A ⊗ | m (cid:105) B in the computational a r X i v : . [ c ond - m a t . d i s - nn ] J un basis | M (cid:105) A , 1 ≤ M ≤ N A , and | m (cid:105) B , 1 ≤ m ≤ N B , ofthe two subsystems A and B , respectively.For fully random states, D = 1, the mean von Neu-mann EE is given by the Page value [22] S Page ( A ) = ln N A − N A N B , (3)and its fluctuations decays to zero, δ S Page ( A ) = ( S ( A ) −S ( A )) / ∼ N − B [43–45]. The overline indicates therandom vector average.Moreover, the ergodic properties of the wave function { ψ n =( M,m ) } can be characterized in terms of multifrac-tal analysis [24] via the fractal dimensions D q , q ≥ q with N , D q ln N = ln IPR q − q , with IPR q = (cid:88) n | ψ n | q , (4)giving in the limit q → D ln N = − (cid:80) n | ψ n | ln | ψ n | .The exponent D provides important information onthe dimension of the support set of the wave functionin the Fock space, which scales as ∼ N D [46]. Ergodicstates are characterized by D q = 1, meaning that thestate is homogeneously spread over the entire Hilbertspace [25]. Instead, NEE states are usually multifrac-tal with D q < ∼ N D /N . log2 L/ S ( L / ) ( a ) D =0 . . . . . S Page D D e n t ( b ) S ( L/ ∼ D ent log2 L/ ExactTheoryDiagonal Approx. D D f l u c FIG. 1. von Neumann EE scaling versus fractal dimen-sion D for S-RPS . (a) S ( L/
2) of half-system, N A = 2 L/ ,versus L for different D ; dashed line shows the Page value,Eq. (3). (b) Slope D ent of S ( L/ ∼ D ent L/ D (Exact) and of − (cid:80) i ρ M,M ln ρ M,M ∼ D ent L/ D fluc of the standard deviation δ S ( L/ ∼ D fluc ln 2 L/ versus D . Black dashed line showsanalytical prediction, Eq. (E2), for p = 1 / In this work, we consider entanglement properties ofNEE states by introducing the sparse random pure states(S-RPS). The S-RPS are normalized random vectors { ψ n } with N D non-zero elements, that are randomly dis-tributed over the Hilbert space. Similarly to RPS [22]( D = 1) all non-zero coefficients are Gaussian distributedwith the normalization-controlled width. The S-RPS are described by only one fractal dimension D q = D < q ∼ N D (1 − q ) , q > Results — We start to outline our results, by comput-ing numerically the mean EE for S-RPS with fractal di-mension 0 < D < N = 2 L [48]. In this case, the S-RPS could be thoughtas eigenstates in the middle of the spectrum [49] of somestrongly interacting spin- chain with L sites.First, we consider limiting cases: for D = 1, S ( A ) isgiven by the Page value, Eq. (3), ∼ ln N A , as the systemis ergodic, while for D = 0, the wave function is localizedin the Fock-space and EE shows area-law S ( A ) ∼ O (1).For 0 < D <
1, one may expect the natural interpolation S ( A ) ∼ D ln N A , as S-RPS are random states in a sub-Hilbert space of dimension N D . However, as we willshow, this intuitive picture is misleading.Figure 1 presents the mean value of the half-partitionEE, S ( L/ N A = 2 L/ . S ( L/
2) follows a volume law S ( L/ ∼ D ent ln 2 L/ for any D > D ent grows with increasing D . However, the curves approachthe Page value S P age ( L/
2) = L/ − / D ent = 1 for D > /
2. Instead, for
D < / S ( L/
2) grows slower than S P age ( L/
2) and wefound D ent = 2 D , Fig. 1 (a)-(b). Thus, basing only onthe mean EE, one might erroneously conclude that thesystem is ergodic for D > /
2, even though the wave-function is confined in an exponentially small ratio ofthe total Hilbert space ∼ − (1 − D ) L .To understand the above phenomenon, we considerthe structure of the reduced density matrix ρ A , Eq. (2),determined by scalar products of the vectors ψ M =( ψ M, , . . . , ψ M,N B ). For M (cid:54) = M (cid:48) , these vectors are in-dependent [50] and the off-diagonal elements of ρ A arealmost negligible for D < ψ M ,which has only N D /N fraction of non-zero elements. In-stead, the diagonal elements of ρ A are given by the normsof the vectors ψ M and cannot be neglected [51].This analysis can be clearly seen in Fig. 2, which shows ρ A , N A = 2 L/ , for a given random configuration ofthe S-RPS. As one can notice ρ A is always nearly di-agonal. Moreover, for D > /
2, an extensive numberof off-diagonal elements become non-zero and the diago-nal ones are homogeneously distributed with amplitude ρ AM,M ∼ − L/ , Fig. 2 (a)-(b). As soon as D is smallerthan 1 /
2, only few off-diagonal elements of ρ A are non-zero, while the distribution of the diagonal ones is bi-modal with ∼ DL non-zero terms, Fig. 2 (c).As the first approximation, the scaling of EE canbe estimated considering only diagonal elements of ρ A , S ( L/ ∼ − (cid:80) i ρ AM,M ln ρ AM,M ∼ D ent L/ D ent = 1 for D < / D ent = 2 D for D ≥ /
2. We further support the validity of this diago-nal approximation in Appendix A. In Fig. 1 (b), we show D ent both from the EE and its diagonal counterpart andfind the perfect match with the above prediction.The diagonal approximation has been used to describe M M ( a ) D = 1 M ( b ) D = 0 . M ( c ) D = 0 . FIG. 2.
Structure of half-system reduced density matrix | ρ M,M (cid:48) A | / max M,M (cid:48) | ρ M,M (cid:48) A | for S-RPS with (a) D = 1, (b) D =0 .
7, and (c) D = 0 . N A = 2 L/ and L = 12. In all panels ρ A is mostly represented by the diagonal elements with almostuniform distribution ρ M,MA ∼ N − A for D > / ∼ DL non-zero nearly uniform elements normalized as ρ M,MA ∼ − DL with the rest being negligibly small. The correspondingEE saturates at the ergodic Page value S ( A ) = S Page ( A ) for D > /
2, while being dominated by 2 DL non-zero elements for D < / S ( A ) (cid:39) − (cid:80) M ρ M,MA ln ρ M,MA ∼ DL ln 2. D D e n t ( a ) S ( L/ ∼ D ent log2 L/ D D e n t ( q ) ( b ) S q ( L/ ∼ D ent log2 L/ q = 1 . . D -5 -3 -1 S P a g e − S ( L / ) L =152127 FIG. 3.
Effect of partition size and scaling of RenyiEE . (a) Slope D ent of the mean EE versus fractal dimen-sion D with N A = 2 L/ . (inset) S Page ( L/ − S ( L/
3) versus D showing the corrections to the Page value exponentiallysuppressed with L for D > p = 1 / D ent ( q ) of the mean Renyi EE of a half-system fordifferent q . The dashed blacks lines represent theoretical pre-dictions in Eq. (7). thermodynamic entropy out-of-equilibrium [52, 53] andit can be analytically verified in terms of leading scal-ing behavior. As only few off-diagonal elements of ρ A are non-zero (say, ρ AM,M (cid:48) for the M th row) one can es-timate the Schmidt eigenvalues λ M and λ M (cid:48) by diago-nalizing the 2 × (cid:18) ρ AM,M ρ AM,M (cid:48) ρ AM (cid:48) ,M ρ AM (cid:48) ,M (cid:48) (cid:19) . Finally, bythe Cauchy-Bunyakovski-Schwarz inequality | ρ AM,M (cid:48) | ≤ ρ AM,M ρ AM (cid:48) ,M (cid:48) , one concludes that the Schmidt eigenval-ues λ M and λ M (cid:48) scale with N as the diagonal elements ρ AM,M , ρ AM (cid:48) ,M (cid:48) (see Appendix C 1). Furthermore, in thisleading approximation the mean EE is given by S ( L/ (cid:39) − (cid:88) M ρ AM,M ln ρ AM,M ∼ ln N , (5) where N is the number of non-zero diagonal elements ρ AM,M = (cid:80) N B m =1 | ψ M,m | [54], which have almost all thesame value (see Fig. 2).The probability distribution P ( N ) of N can be com-puted combinatorically. Let g M be the number of non-zero elements giving contributions to ρ AM,M . By construc-tion of the S-RPS we have (cid:80) M g M = N D . Now, P ( N )is proportional to the product of the number of combi-nations (cid:16) N D − N − (cid:17) to realize N non-zero g M > (cid:16) N A N (cid:17) to place them among N A values of 1 ≤ M ≤ N A . The typical N is given by theposition of the maximum of its probability distribution N typ0 = N A N D N A + N D (cid:39) N min( p,D ) , (6)confirming the numerical result, Fig. 1, S ( A ) (cid:39) (cid:26) D ln N, D < p ln N A , D > p . (7)Importantly, the S-RPS do not have any intrinsic lo-cality due to randomly-chosen positions of the non-zeroelements. Thus, Eq. (7) gives a natural upper boundfor the maximal EE for generic many-body/multifractalwave functions with support set ∼ N D [55].Now, we further numerically test our main result,Eq. (7), by computing S ( A ) for a different subsys-tem A . Figure 3 (a) shows the slope D ent of S ( N A ) ∼ D ent ln N A for N A = 2 L/ as a function of the frac-tal dimension D . For D > /
3, we have D ent = 1and EE shows ergodic behavior. For smaller D , D ent deviates from the infinite temperature thermal value, D ent = 3 D , in agreement with Eq. (7). The difference S Page ( L/ − S ( A ) is shown in the inset in Fig. 3 (a) -2 -1 b ( a ) PLBM D D D ent D D e n t ( b ) L = 81012 Theory
FIG. 4.
Scaling exponents of EE, D ent , Shannon en-tropy, D , and IPR, D for PLRBM. (a) D ent , D , and D extracted by linear extrapolation from EE, Shannon en-tropy D ln N , and the IPR D ln N scalings versus subsystemsize N A = 2 L/ .(b) The parametric plot of D ent versus D .Different curves correspond to the different points L − L , L + 2 of an enlarging linear fitting procedure. supporting the convergence of EE to the Page value S Page ( L/
3) up to exponentially small corrections in L (as well as S Page ( L/
2) in Fig. 1 (a)).Furthermore, our results can be generalized also forthe Renyi EE, Eq. (1), and for genuine multifractalstates (see Appendix F). Figure 3 (b) shows S q ( A ) ∼ D ent ( q ) ln N A with N A = 2 L/ for several q >
1. In agree-ment with Eq. (7), we obtain D ent ( q ) = 1 for D > / D ent ( q ) = 2 D otherwise. The q -independence of D ent ( q ) at q ≥ D q = D for q > D q , and for N A = N p , the upper bound,Eq. (7) rewritten as the lower bound for fractal dimen-sions D q ln N ≥ S q ( A ), q ≥
1, of a state with fixedRenyi entropies 0 ≤ S q ( A ) ≤ ln N A can be proved tobe strict and can be saturated by the change of thesubsystem bases [55]. Indeed, one can show [47] thatthe minimal value D q can be achieved if the computa-tional basis is optimized to be the Schmidt decomposi-tion basis | ψ (cid:105) = (cid:80) a λ / M a | M a (cid:105) A ⊗ | m a (cid:105) B as in this case ψ M a ,m a (cid:48) = λ / M a δ a,a (cid:48) , see Appendix F.In order to demonstrate the validity of this generalbound in Fig. 12, we show the scaling of the fractal ex-ponents, D and D extracted from IPR q , and D ent fromEE, for the paradigmatic example of power-law randombanded matrices (PLBM) [24] known to have genuinemultifractality of eigestates at the critical point, tunedby the parameter b . Plotting D ent versus D , we see thatthe universal bound is satisfied, though not saturated forthe inspected system sizes. We show the results for an-other exemplary many-body system in Appendix G. Fluctuations — Quantum fluctuations represent an-other important ingredient to understand ergodicity. Ac-cording to ETH, they can be related to temporal fluctua-tions around the equilibrium value in a quench protocol.In ergodic systems the scaling of fluctuations is relatedto the dimension of the larger subsystem playing the role of a bath [43–45, 56]. EE fluctuations can be quantifiedby the standard deviation δ S ( A ) = ( S ( A ) − S ( A )) / ∼ N − D fluc / , (8)from the collapse with L of the probability distribution P ( x ) of the rescaled variable x = ( S ( A ) − S ( A )) /δ S ( A )(see Appendix E). Importantly, the scaling of fluctuationsdisplays three different regimes for a generic cut N A = N p , p ≤ /
2, (see inset in Fig. 2 for p = 1 / D fluc = D, D < p D − p, p < D < − p/ − p ) , D > − p/ . (9)For D < p , both mean EE and its fluctuations show theproperties of local observables: their scaling is related tothe equilibration within the fractal support set N D anddoes not depend on the subsystem size. For p < D < − p/ δ S ( A ) ∼ N − (2 D − p ) . Finally for 1 − p/ < D <
1, both mean and its fluctuations are undistinguishablefrom ergodic states at infinite temperature [57].
Conclusions — In order to answer to the main questionof the paper – to what extend ergodicity properties ex-tracted from entanglement and multifractal probes pro-vide the same description of thermal phases – we general-ized the result of Ref. [22] on the EE for RPS to the caseof NEE states characterized by fractal dimensions D q . Inparticular, we presented an upper bound for the entan-glement entropies S q (both von Neumann and Renyi) ofa generic multifractal state with fractal dimensions D q ,see Appendix F.This bound leaves the gap for S q ( A ) to be equal tothe Page value provided the wave function support set islarger than the subsystem size, N D q > N A . An exampleof the saturation of this bound is shown for a newly in-troduced class of sparse random pure states. Our resultsshow that for small fractal dimensions N D q < N A EEbehaves as a local observable both in terms of the meanvalue and fluctuations.Thus, ergodicity viewed as the wave function equipar-tition in the full Hilbert space is more strict than the oneimposed by the value of the EE.Our results find immediate application in the many-body localization theory where EE is used to probe thetransition, or in strongly kinematically constrained mod-els where ergodicity may break down due to Fock/Hilbertspace fragmentation. For instance, in spin models inRefs. [58, 59], the eigenstates live on an exponentiallysmall fraction of the full Hilbert, due to dipole conserva-tion [58, 60] and strong interactions [59] (Fock-space frag-mentation). Nevertheless, the half-chain entanglemententropy equals to the Page value, provided the fractaldimension of the wave function support set is
D > / ACKNOWLEDGMENTS
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B , 7920–7933 (2000).[92] A D Mirlin and Y V Fyodorov, “The statistics of eigen-vector components of random band matrices: analyticalresults,” J. Phys. A: Mathematical and General , L551(1993). Appendix A: Numerical tests of diagonalapproximation
In this section, we provide further numerical evidenceof the validity of the diagonal approximation. In the maintext we used the diagonal approximation S diagonal1 ( A ) = − (cid:80) M ρ AM,M ln ρ AM,M to estimate the scaling of the vonNeumann EE S ( A ). Figure 5 show both S ( A ) and S diagonal1 ( A ) for half-partition N A = 2 L/ , as functionof L for several fractal dimension D , giving indicationthat S ( A ) (cid:39) S diagonal1 ( A ).To understand numerically why the diagonal approxi-mation works well, we analyze in more detail the struc-ture of the reduced density matrix ρ A . First, we start toinvestigate the sparse property of ρ A . For this purpose,we define the sparsity of ρ A as the number of its non-zero ( a ) D = 0 . S ( L/ − X M ρ L/ M,M log ρ L/ M,M ( b ) D = 0 . ( c ) D = 0 . log2 L/ ( d ) D = 0 . log2 L/ ( e ) D = 0 . log2 L/ ( f ) D = 0 . FIG. 5.
Diagonal approximation and von Neumann EE (a)-(f) half-partition averaged EE S ( L/
2) (solid-line) and itsdiagonal approximation S diagonal1 ( L/
2) = − (cid:80) M ρ AM,M ln ρ AM,M (dashed line) as a function of L for several fractal dimension D . L/ (2 L/ − s p a r s i t y ( a ) D =0 . . . . . . . . L/ (2 L/ − D D s ( b ) sparsity ∼ D s L FIG. 6.
Sparsity of the off-diagonal elements of ρ A for N A = 2 L/ . (a) Sparsity S defined as the number of non-zerooff-diagonals in the density matrix as a function of the totalnumber N A ( N A −
1) of off-diagonals of ρ A at half-partition( N A = 2 L/ ) for several D . (b) Rate D s of non-zero off-diagonal elements, S ∼ pD s L as a function of the fractaldimension D . off-diagonal elements, S = {| ρ M,M (cid:48) A | (cid:54) = 0 } . Figure 6(a)and Fig. 7(a) show the sparsity of ρ A for two differentpartition N A = 2 L/ and N A = 2 L/ , respectively. Asone can notice, for large D the number of non-zeros in ρ A grows as the total size of the reduced density matrix N A meaning that the matrix is not sparse for these val-ues of D . Nevertheless, for small D , ρ A is sparse and thenumber of non-zeros elements of ρ A is an exponentially L/ (2 L/ − s p a r s i t y ( a ) D =0 . . . . . . . . D D s ( b ) sparsity ∼ D s L/ FIG. 7.
Sparsity of the off-diagonal elements of ρ A for N A = 2 L/ . (a) Sparsity as function of N A ( N A −
1) forseveral D . (b) Rate D s of non-zero off-diagonal elements, S ∼ pD s L/ as a function of the fractal dimension D . small fraction of the full dimension.To better quantify the sparsity of ρ A , we define therate D s as S ∼ N D s p . For D s = 1 the matrix is notsparse, and ρ A is diagonal for D s = 0. Figure 6(b) andFig. 7(b) show D s for two different partitions N A = 2 L/ and N A = 2 L/ , respectively. As expected, for large D wehave D s = 1, while D s is proportional to D for smaller D .In the next section, we will give an analytical argumentshowing D s (cid:39) (cid:26) D − p p , D < p , D > p . (A1)and demonstrate that sparsity plays a major role for thevalidity of the diagonal approximation. log2 L/ -8 -7 -6 -5 -4 -3 -2 -1 | ρ A M , M | ( a ) D =0 . . . . . . . . . D D o ff ( b ) | ρ AM,M | ∼ − D off L/ FIG. 8.
Mean off-diagonal element of ρ A for N A =2 L/ . (a) | ρ AM,M (cid:48) | as a function of L for several D . (b) D off exponent extracted from | ρ AM,M (cid:48) | ∼ − pD off L , p = 1 / D . log2 L/ -7 -6 -5 -4 -3 -2 -1 | ρ A M , M | ( a ) D =0 . . . . . . . D D o ff ( b ) | ρ AM,M |∼ − D off L/ FIG. 9.
Mean off-diagonal element of ρ A for N A =2 L/ . (a) | ρ AM,M (cid:48) | as a function of L for several D . (b) D off exponent extracted from | ρ AM,M (cid:48) | ∼ − pD off L , p = 1 / D . Now, we calculate the mean off-diagonal elements of ρ A (not only non-zero ones). Figure 8 (a) and Fig. 9 (a) show | ρ AM,M (cid:48) | as function of L for several D for two differentpartitions N A = 2 L/ and N A = 2 L/ , respectively. Ingeneral, we have | ρ AM,M (cid:48) | ∼ N − D off p . Figure 8 (b) andFig. 9 (b) show D off as a function of D . In the nextsection, we will show that pD off (cid:39) (cid:26) p − D, D < p p , D > p . (A2) Appendix B: Structure of reduced density matrix
In this section, we consider the structure of diagonal ρ AM,M = N B (cid:88) m =1 | ψ M,m | , (B1)and off-diagonal ρ AM,M (cid:48) = N B (cid:88) m =1 ψ M,m ψ ∗ M (cid:48) ,m , (B2)elements of the reduced density matrix ρ A assuming thevectors ψ M and ψ M (cid:48) to be uncorrelated for M (cid:54) = M (cid:48) witha certain probability distribution of each element P ( ψ M,m ) = (1 − p ) δ ( ψ M,m )+ p P ( N D/ ψ M,m ) N D/ . (B3)Here, p = N D /N is the probability that ψ M,m (cid:54) = 0. P ( y ) is the probability distribution of non-zero values,which is symmetric P ( − y ) = P ( y ), has a unit variance (cid:82) y P ( y ) dy = 1 and the fourth cumulant σ = (cid:82) ( y − P ( y ) dy ∼ O (1) . The latter conditions ensure thescaling | ψ M,m | ∼ N − D of non-zero elements and thewave function normalization (on average). In the limit oflarge N , we can further neglect the correlations relatedto the normalization condition.Next, within the above assumptions one can find theprobability distributions of diagonal, Eq. (B1), and off-diagonal, Eq. (B2), elements of the reduced densitymatrix (similar to [62]). For this purpose we rewriteEq. (B3) in a short form for N D/ ψ M,m = yP ( y ) = (1 − p ) δ ( y ) + p P ( y ) . (B4)
1. Probability distribution of diagonals ρ AM,M
Here we use the Fourier transform to calculate the N B -fold convolution of the probability distribution ˜ P ( t (cid:48) ) = P ( √ t (cid:48) ) √ t (cid:48) of t (cid:48) = | ψ M,m | and obtain P ( ρ AM,M ) = N B (cid:88) k =0 (cid:18) N B k (cid:19) (1 − p ) N B − k p k ˜ P k ( ρ AM,M N D ) , (B5)with˜ P k ( t ) = 12 π (cid:90) e − iωt (cid:90) P (cid:16) √ t (cid:48) (cid:17) √ t (cid:48) e iωt (cid:48) k dω . (B6)The scaling of p = N D − and N B = N − p providethe optimal index k ∗ = N B p = N D − p (B7)0giving the main contribution to the sum Eq. (B5).As k is integer, one has to distinguish two cases:(i) D < p when k ∗ = N B p (cid:28) P ( ρ AM,M = x ) dx (cid:39) (1 − k ∗ ) δ ( x ) dx + k ∗ ˜ P ( N D x ) N D dx , (B8)and (ii) D > p when k ∗ = N B p (cid:29) P ( ρ AM,M ) = e − ( ρ AM,M − N − p ) / (2 σ N − D − p ) √ σ N − D − p . (B9)This analysis shows that for D > p the diagonal ρ A -elements are homogeneously distributed with the meanvalue ρ AM,M = 1 /N A given by Tr[ ρ A ] = 1.
2. Probability distribution of off-diagonals ρ AM,M (cid:48)
To obtain P ( ρ AM,M (cid:48) ) one has to calculate, first, fromEq. (B4) P ( N D ψψ (cid:48) = z ) = (cid:120) P ( y ) P ( y (cid:48) ) δ ( z − yy (cid:48) ) dydy (cid:48) = (1 − p ) δ ( z ) + p ¯ P ( z ) , (B10)with ¯ P ( z ) = (cid:120) P ( y ) P ( y (cid:48) ) δ ( z − yy (cid:48) ) dydy (cid:48) . (B11)Then, analogously to the previous subsection, one canuse the Fourier transform to calculate P ( ρ AM,M (cid:48) ) = N B (cid:88) l =0 (cid:18) N B l (cid:19) (1 − p ) N B − l p l ¯ P l ( ρ AM,M N D ) , (B12)with ¯ P l ( t ) = 12 π (cid:90) e − iωt (cid:18)(cid:90) ¯ P ( z (cid:48) ) e iωz (cid:48) (cid:19) l dω . (B13)The scaling of p = N D − and N B = N − p providethe optimal index l ∗ = N B p = N D − − p (B14)giving the main contribution to the sum Eq. (B12).As l is integer, one has to distinguish two cases: (i) D < p when l ∗ = N B p (cid:28) P ( ρ AM,M (cid:48) = x ) dx (cid:39) (1 − l ∗ ) δ ( x ) dx + l ∗ ¯ P ( N D x ) N D dx , (B15)and (ii) D > p when l ∗ = N B p (cid:29) P ( ρ AM,M (cid:48) ) = e − ( ρ AM,M (cid:48) ) / (2 σ N − − p ) √ σ N − − p . (B16) Here we used the fact that ¯ P ( z ) = ¯ P ( − z ) is symmetricand thus there is no drift in CLT.The latter analysis confirms the scaling of the off-diagonal elements, Eq. (A2), as well as the number ofnon-zero off-diagonals, Eq. (A1). Indeed, for D > p the distribution is smooth with the typical value ρ AM,M (cid:48) ∼ N − p , (B17)thus, the rate of non-zero off-diagonal elements D s = 1and their scaling | ρ AM,M (cid:48) | ∼ − pD off L is D off = p p .In the opposite limit of D < p the distribution isbimodal giving the number of non-zeros N pD s = N A l ∗ = N D − p (B18)as well as the mean value | ρ AM,M (cid:48) | = N − pD off = l ∗ N − D = N D − − p . (B19) Appendix C: Sparseness of the reduced densitymatrix for non-ergodic states
Now, we provide an analytical argument to supportthe validity of the diagonal approximation in the regimein which ρ A is sparse. As we are interested in the scalingof the Schmidt values with N compared to the one ofdiagonal elements ρ AM,M , we have to consider two cases:(i) First, when the number of non-zero elements in eachrow is finite and does not grow with N , the off-diagonalelements can be of the same order as the diagonal ones.(ii) Second, when there are many non-zero off-diagonalswhich are much smaller than ρ AM,M .
1. Few non-zero off-diagonal elements ρ AM,M (cid:48) ,( D < / ) As follows from Eq. (A1) there is at most O (1) non-zero off-diagonal elements in each row as soon as D < / ∼ N A ).In this case, we can show that in terms of multifractalscaling with the total Hilbert space dimension N in theabove regime the Schmidt values λ M scale in the sameway as the diagonal elements of ρ A and, thus, EE can beapproximated by its diagonal counterpart [52, 53] S q ( p ) = ln Σ q − q , Σ q = (cid:88) M λ qM (cid:39) (cid:88) M (cid:0) ρ AM,M (cid:1) q . (C1)Indeed, if in each row of ρ A there are only few signif-icantly non-zero off-diagonal matrix elements (say, for M th and M (cid:48) th diagonals), then Schmidt eigenvalues canbe approximated by diagonalizing a 2-by-2 matrix (cid:18) ρ AM,M ρ AM,M (cid:48) ρ AM (cid:48) ,M ρ AM (cid:48) ,M (cid:48) (cid:19) . (C2)1Assuming the following scaling ρ AM,M ∼ N − α M , and ρ AM,M (cid:48) ∼ N − β , with α M ≤ α M (cid:48) without loss of gener-ality, we obtain for the corresponding Schmidt values λ M/M (cid:48) = N − α M + N − α M (cid:48) ± (cid:113) ( N − α M + N − α M (cid:48) ) + 4 N − β (cid:39) (cid:26) N − α M + N − β + α M (cid:39) N − α M N − α M (cid:48) + N − β + α M (cid:39) N − α M (cid:48) . (C3)The latter approximation is based on the inequality β ≥ ( α M + α M (cid:48) ) / (cid:12)(cid:12) ρ AM,M (cid:48) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N B (cid:88) m =1 ψ M,m ψ ∗ M (cid:48) ,m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:118)(cid:117)(cid:117)(cid:116) N B (cid:88) m =1 | ψ M,m | N B (cid:88) m (cid:48) =0 | ψ M (cid:48) ,m (cid:48) | = (cid:113) ρ AM,M ρ AM (cid:48) ,M (cid:48) . (C4)As a result, the scaling of Schmidt values λ M with N isshown to be the same as for the diagonal elements ρ AM,M in the nearly diagonal sparse regime of ρ A ( D < /
2. Many non-zero off-diagonal elements ρ AM,M (cid:48) ,( D > / ) In the case of
D > / D = 1 considered in [45].In the case of D > p > p both diagonal andoff-diagonal elements of the reduced density matrix arehomogeneously distributed and the latter has the formsimilar to the Rosenzweig-Porter random matrix ensem-ble [80]. Then the Schmidt eigenspectrum is not affectedby the off-diagonal elements when [81] | ρ AM,M (cid:48) | ρ AM,M ∼ N − − p (cid:28) N − p/ , (C5)which is the case as p ≤ / D > / N A × N A matrix, N A = N p , as follows: the spec-trum is not affected by the off-diagonal elements as soonas N p | ρ AM,M (cid:48) | | ρ AM,M | (cid:28) . (C6) In our case it leads to N p | ρ AM,M (cid:48) | | ρ AM,M | ∼ (cid:26) N D − p , D < p N p − , D > p , (C7)and works for any 0 ≤ D ≤ Appendix D: Entanglement entropy for fractal states
In this section, we consider mean and fluctuations ofRenyi and von Neumann EE within the approximationsof two previous sections.The simplest way to calculate the Renyi entropy,Eq. (C1), in the diagonal approximation λ M (cid:39) ρ AM,M = N B (cid:88) m =1 | ψ M,m | (D1)is to use the probability distributions, Eq. (B8), and,Eq. (B9). Indeed,Σ q = N A (cid:16) ρ AM,M (cid:17) q = (cid:26) N D (1 − q ) , D < pN p (1 − q ) , D > p , (D2)leading straightforwardly to Eq. (8) of the main text.The fluctuations can be also estimates from the mo-ments as soon as the variance σ = Σ q − Σ q = N A (cid:34)(cid:16) ρ AM,M (cid:17) q − (cid:16) ρ AM,M (cid:17) q (cid:35) (D3)is small compared to Σ q . Indeed, as(1 − q ) S q ( p ) = ln Σ q = ln Σ q + ln (cid:20) σ Σ Σ q g q (cid:21) (cid:39) ln Σ q + σ Σ Σ q g q − σ q g q (D4)it gives (1 − q ) S q ( p ) = ln Σ q − σ q (cid:39) ln Σ q (D5)within the leading approximation, and(1 − q ) (cid:104) S q ( p ) − S q ( p ) (cid:105) = σ Σ q . (D6)Here we introduced dimensionless variable g q = Σ q − Σ q σ Σ with zero mean and unit variance g q = 0 , g q = 1 . (D7)In our case one obtains σ Σ q = N − D , (D8)giving the correct approximation for D < p .2
1. Alternative way to calculate entanglemententropies
Alternatively in the main text we parameterizeSchmidt values as follows λ M (cid:39) ρ AM,M = N B (cid:88) m =1 | ψ M,m | = g M /N D , (D9)where 0 ≤ g M ≤ N D are integer values summed to thesupport set N D : N A (cid:88) M =1 g M = N D . (D10)The entanglement entropy in this case can be esti-mated as the logarithm of the number N of non-zero g M S ∼ ln N . (D11)As we show below, this approximation is good for meanEE for any D , but fails to capture fluctuations for D > p .The probability distribution P N of N can be calcu-lated combinatorically in the assumption of homogeneousdistribution of g M ’s. Indeed, the total number of com-binations of N A values of g M , 1 ≤ M ≤ N A , taken withrepetitions ( g M can be larger than 1) and with the nor-malization Eq. (D10) is given by M = (cid:18) N A + N D − N D (cid:19) . (D12)At the same time the combinations with N non-zero g M can be counted as the number of combination to realize N non-zeros M ¯0 = (cid:18) N D − N − (cid:19) (D13)times the number of combinations to place them among N A , which is M N = (cid:18) N A N (cid:19) . (D14)As a result P N = M ¯0 M N M (cid:39) A ( N ) e N A f ( ρ ) , (D15)where f ( ρ, α ) = − ρ ln( ρ ) − ( α − ρ ) ln( α − ρ ) − (1 − ρ ) ln(1 − ρ ) , (D16) N D = αN A , and 0 ≤ ρ = N /N A ≤ , α and we ne-glected − N D and N . The expres-sion for f ( ρ ) is calculated in the large- N limit with helpof Stirling’s approximation. The maximum of f ( ρ ) is achieved at the typical N ∗ = N A ρ ∗ with ρ ∗ = α α < , α leading to N ∗ = N A N D N A + N D (D17)from the main text.The relative fluctuations δN /N ∗ = δρ/ρ ∗ can be writ-ten in the following form δN N ∗ = δρ ρ ∗ = 1 (cid:112) − N A f (cid:48)(cid:48) ( ρ ∗ ) ρ ∗ = (cid:0) N A + N D (cid:1) − / (D18)in the Gaussian approximation P N = e − ( N A + N D )( ρ − ρ ∗ ) / (cid:112) π/ ( N A + N D ) , (D19)derived from Eq. (D15) and Eq. (D16) provided ρ ∗ (cid:29) ( N A + N D ) − / .In the same approximationln N = ln N ∗ − N A + N D ) , (D20a)ln N = ln N ∗ + 1 − ln N ∗ ( N A + N D ) . (D20b)According to Eq. (D11) and Eq. (D20a) mean EE isgiven by S (cid:39) ln N = ln N ∗ − N D + N A ) ∼ (cid:26) ln N D = D ln N, for D < p ln N A = p ln N, for D > p . (D21)In the latter equality we neglected subleading terms.At the same time according to Eq. (D20a) andEq. (D20b) EE fluctuations are given mostly by the rel-ative fluctuations of N S − S (cid:39) ln ( N ) − ln( N ) (cid:39) N D + N A . (D22)As mentioned above the approximation Eq. (D11)works both for the mean and fluctuations provided D
p when N (cid:39) N A (cid:28) N D there is a non-trivial distribution of g M with g M = N D /N A (cid:29) N (cid:54) = ln (cid:2)(cid:80) M (cid:0) g M N D (cid:1) q (cid:3) − q . (D25)Nevertheless, as we have shown in Sec. A, on average bothsides of the latter equation give the same Page value asEq. (8) in the main text. Appendix E: Collapse of the EE probabilitydistribution for S-RPS
Here, we numerically characterize EE fluctuations forthe S-RPS. EE fluctuations can be quantified by the stan-dard deviation δ S ( A ) = ( S ( A ) − S ( A )) / ∼ N − D fluc / , (E1)from the collapse with L of the probability distribution P ( x ) of the rescaled variable x = ( S ( A ) − S ( A )) /δ S ( A ).Figure 10 shows the collapse of P ( x ) with L for S-RPSfor several D and N A = 2 L/ . Scaling of fluctuationsextracted from the above collapse displays three differentkinds of behavior for a generic cut N A = N p , p ≤ / p = 1 / D fluc = D, D < p D − p, p < D < − p/ − p ) , D > − p/ . (E2)For D < p , both mean EE and its fluctuations show theproperties of local observables: their scaling is relatedto the equilibration within the fractal support set N D and does not depend on the subsystem size – and cor-respond to the analytical results of the previous section.For p < D < − p/ δ S ( A ) ∼ N − (2 D − p ) .Finally for 1 − p/ < D <
1, both mean and its fluctua-tions are undistinguishable from ergodic states at infinitetemperature.
Appendix F: Upper bound for the meanentanglement entropy for multifractal states
For genuine multifractal states the above mentionedderivations are not applicable as its probability distribu-tion is not bimodal like in Eq. (B3) or Eq. (B4). Thus, inorder to proceed we use another approach and show thatthere is a generic upper bound for the mean Renyi en-tanglement entropy S q ( A ) for a state with certain fractaldimensions 0 ≤ D q ≤ S q ( A ) ≤ (cid:26) D q ln N, D q < p ln N A , D q > p (F1) or vice-a-versa there is a generic lower bound for the frac-tal dimensions D q for a state with certain fixed Renyientropies 0 ≤ S q ( A ) ≤ ln N A D q ≥ S q ( A )ln N . (F2)Here and further, we restrict our consideration to thephysically relevant moments q ≥ S q ( A ), but not D q ) or by basis transformations of sub-systems (which changes only D q , but not S q ( A )).As both Eq. (F1) and (F2) are equivalent, let’s provethe second one. For this we consider the state | ψ (cid:105) = (cid:88) a λ / α | E α (cid:105) A ⊗ | ε α (cid:105) B (F3)written in the Schmidt decomposition basis, where λ α are (non-zero) Schmidt eigenvalues and | E α (cid:105) A and | ε α (cid:105) B are the corresponding eigenvectors in subsystems A andB. From Eq. (C1) the q th Renyi entropy is given by S q ( A ) = ln [ (cid:80) M λ qM ]1 − q (F4)and does not depend on the eigenvectors | E α (cid:105) A and | ε α (cid:105) B .On the other hand, the coefficients ψ M,m of | ψ (cid:105) in thecomputational basis | M (cid:105) A ⊗ | m (cid:105) B are highly sensitive tothese vectors ψ M,m = (cid:88) a λ / α (cid:104) M | E α (cid:105) A (cid:104) m | ε α (cid:105) B , (F5)so are the fractal dimensions D q , Eq. (4)IPR q ≡ N − D q ( q − = (cid:88) M,m | ψ M,m | q . (F6)The minimization of D q is equivalent to the maximiza-tion of the corresponding IPR. Due to q > | ψ M,m | by their squared sum due to the general-ized Cauchy-Bunyakovski-Schwarz inequality a q + b q < ( a + b ) q , a, b > . (F7)This process is equivalent to the change of the compu-tational basis with respect to the Schmidt basis and islimited by the fixed Schmidt eigenvalues.The maximal value of the IPR q corresponds to thechoice of the computational basis to be the same as theSchmidt eigenbasis | M (cid:105) = | E M (cid:105) , | m (cid:105) = | ε m (cid:105) . (F8)and leads to the diagonal reduced density matrix.4 x = ( S ( L/ − S ( L/ · DL/ S ( L/ ∼ DL log2 D = 0 . ( c ) x = ( S ( L/ − S ( L/ · (2 D − / L/ ( b ) S ( L/ ∼ L/ D = 0 . x = ( S ( L/ − S ( L/ · L/ P ( x ) ( a ) S ( L/ ∼ L/ D = 0 . L = 162024 D D f l u c FIG. 10.
Collapse of the probability distribution P ( x ) of the half-system EE at finite sizes L , x = ( S − S ) /δ S ,for different D : (a) D = 0 .
8, (b) D = 0 .
7, (c) D = 0 .
25. Scaling of the mean EE, S ( A ), is shown in the legend, while thedistribution width scaling is specified in the axis label. (inset) Slope D fluc of the standard deviation δ S ( L/ ∼ D fluc ln 2 L/ versus D . Black dashed line shows analytical prediction, Eq. (E2), for p = 1 / The latter gives immediately ψ M,m = λ / M δ M,m (F9)and N − D q ( q − = (cid:88) M,m | ψ M,m | q = (cid:88) M λ qM = e −S q ( q − . (F10)This concludes the proof of the inequality (F2) as in allsteps we have just increased the IPR, i.e., decreased D q .Thus, in a general case of a genuine many-body mul-tifractal state, the inequalities (F1) and (F2) are valid,however their saturation to equalities is an open and dif-ficult question needed further investigations.In the main text we have provided an example of S-RPS saturating this bound. Here as a counterexamplelet’s consider fractal NEE states with the structure of thewave function forming a so-called ergodic bubble [83–88].Let’s consider the states with only N D non-zero elements,which are Gaussian distributed with the width controlledby the normalization, but unlike S-RPS the distributionof these coefficients will be different. We associate eachconfiguration among N = 2 L ones with the state of L spins-1 / DL to be an integer number. Thenby the ergodic-bubble structure we would mean that forthe randomly chosen DL spins from L the wave-functioncoefficients are non-zero for all the 2 DL configurations ofthe selected spins, while the rest (1 − D ) L spins are frozenin a fixed state.It is easy to show that as soon as DL spins are chosenrandomly among the subsystems A (with pL spins) andB (with (1 − p ) L spins) the fraction of the chosen spinsin the subsystem A is the same as in the total systemand equal to D . As a result, the reduced density matrixin the subsystem A is represented by the product stateof ∼ (1 − D ) pL spins and an ergodic bubble of the size ∼ DpL . The resulting entanglement entropy is equal to S bubble q ( A ) = D ln N A , (F11)which saturates the bound (F1) only in the trivial cases D = 0 and D = 1. Appendix G: Further numerical examples
In this section we provide two additional examples ofthe random matrix and many-body models confirmingthe general results for the upper bound of EE, Eq. (8) ofthe main text, and the possibility of EE to be saturatedat the Page value for strictly non-ergodic wave functionswith D <
1. Power-law random banded matrix
In this section, we focus on entanglement properties ofthe power-law random banded matrix (PLRBM) ensem-ble [24] and their relations with the fractal exponents.The PLRBM are defined by (cid:98) H = L (cid:88) x,y h x,y | x (cid:105)(cid:104) y | , (G1)where h x,y = µ x,y / (1 + ( | x − y | /b ) α ) / and µ x,y = µ y,x are independent random variables uniformly distributedin the interval [ − , (cid:98) H are multifractal, therefore we fixthe parameter α to its critical value at the Andersonlocalization transition (ALT) α = 1 [89]. The parameter b tunes the multifractal properties of the PLRBM. For b (cid:29) L log2 − l og ( I P R ) ( c ) L/ S ( a ) S Page = L/ − / L log2 I P R ( b ) FIG. 11.
Scaling of EE, Shannon entropy D ln N , and the logarithm of IPR D ln N as a function of subsystemsize N A = 2 L/ for PLRBM at the critical point, α = 1 for different multifractal parameter b . (a) Scaling ofthe mean von Neumann entanglement entropy S ( L/
2) of half-system, N A = N / = 2 L/ , as a function of L . Dashed linestands for the Page value scaling; (b) The mean Shannon entropy D ln N versus L ; (c) The mean IPR fractal dimension D ln N = ln IP R versus L . Values of the multifractal parameter b for all three panels are shown in the legend of panel (c). -3 -2 -1 b ( a ) D D D ent D D e n t ( b ) L = 81012 Theory
FIG. 12.
Scaling exponents of EE, D ent , Shannon en-tropy, D , and IPR, D . (a) D ent , D , and D extractedfrom Fig. 11 by linear extrapolation. (b) The parametric plotof D ent versus D . Different curves correspond to the dif-ferent points L − L , L + 2 of an enlarging linear fittingprocedure. − D ∝ b − and for b (cid:28) D ∝ b [24, 90–92]. (cid:98) H is defined in aHilbert space of the dimension 2 L , thus the states | x (cid:105) could be represented in terms of spin-1 / S with the subsystem Hilbert space dimension N A =2 L/ as a function of L for several b . We averaged S overrandom configurations, few eigenstates in the middle ofthe spectrum of (cid:98) H and random partitions. As expectedfor large b , we have S (cid:39) S
P age = L/ − /
2. Asthe parameter b decreases, S scales still with a volumelaw, but for b (cid:46) . S P age , meaning
S ∼ D ent L/ D ent <
1. Figures 11 (b)-(c)show the fractal dimensions D and D correspondingto the Shannon entropy and the logarithm of the IPRversus L for several b . Usually, the computation of fractal exponents D q is a challenging task and in principle theycould have severe finite-size effects (see, e.g., hot debateson the existence of the non-ergodic extended phase in theAnderson model on the random regular graph).We estimate the three exponents D ent , D and D bya linear fit repeating Fig. 4 of the main text as Fig. 12.First, Fig. 12 (a) gives numerical evidence of the existenceof a genuine multifractal phase, meaning that D q is anon-trivial function of q as D < D . Second, as one cannotice, there is a regime in b for which D ent ≈
1, eventough D <
1. To understand the finite-size effects for D ent and D , we use an enlarging linear fitting procedure.This means that we compute D and D ent fitting threeconsecutive system sizes { L − , L, L + 2 } . Figure 12 (b)shows the result of such fitting D ent as function of D forseveral L . As expected, D ent obeys the upper-bound thatwe provided in the main text (dashed line in Fig. 12 (b)).Moreover, the finite-size flow direction suggests that theupper-bound might be saturated in the thermodynamiclimit ( L → ∞ ).
2. A quantum many-body system
In conclusion on the main text, we claimed that theresults for D ent find an immediate applications for quan-tum system where Fock/Hilbert space fragmentation takeplace. In this section, we will give a numerical examplefor the later model.We study the t − V disordered chain of spinless fermionswith periodic boundary conditions, (cid:98) H = − t (cid:88) x (cid:98) c † x +1 (cid:98) c x + h.c. + W (cid:88) x µ x (cid:98) n x + V (cid:88) x (cid:98) n x (cid:98) n x +1 , (G2)where (cid:98) c † x ( (cid:98) c x ) is the fermionic creation (annihilation) op-erator at site x , (cid:98) n x = (cid:98) c † x (cid:98) c x , and µ x are independent6 -5 -4 -3 -2 -1 I P R ( a ) L = 12162024 S ( b ) S page = 1 / N − W − l og I P R / l og N ( c ) W S / S P a g e ( d ) FIG. 13.
IPR and EE for the many-body model withHilbert space fragmentation as functions of disor-der amplitude W . (a) IPR versus W for different sys-tem sizes (shown in legend); (b) EE S versus W for thesame system sizes. Dashed horizontal lines show the Pagevalue for the corresponding system size; (c) Finite-size esti-mate − log IP R/ log N of the fractal dimension D versus W ;(d) Finite-size flow of the S / S Page . In panels (c) and (d) ver-tical dashed lines correspond to the D ( L max ) = 0 .
5, where L max = 24 is the maximal system size we calculated. random variables uniformly distributed in the interval[ − , t = 1 / W and V are the hopping, disorderand interaction strengths respectively. L is the num-ber of sites and we consider the system at half-filling,i.e. the number of particles is n = (cid:80) x n x = L/
2. Thismodel is equivalent to the disordered XXZ Heisenbergspin-1 / V corresponds tothe anisotropy along and perpendicular to the z -axis.At finite interaction strength, V = 1, this model is be-lieved to have a many-body localization (MBL) transitionat W c ≈ . W < W c ergodic and W > W c localized).In our case we consider another limit of large interactionstrengths, i.e. V /t → ∞ . As shown in [59], (cid:98) H can be mapped to the following local Hamiltonian (cid:98) H ∞ = − t (cid:88) x (cid:98) P x (cid:0)(cid:98) c † x +1 (cid:98) c x + h.c. (cid:1) (cid:98) P x + W (cid:88) x µ x (cid:98) n x , (G3)with the dynamical constraints imposed by local projec-tors (cid:98) P x = 1 − ( (cid:98) n x +2 − (cid:98) n x − ) , (cid:98) P x = (cid:98) P x . (G4)In this limit, (cid:80) x (cid:98) n x (cid:98) n x +1 is a new conserved quantity,and the Hilbert/Fock space fragments in several disjointblocks given by the value of (cid:80) x (cid:98) n x (cid:98) n x +1 . We consider thelargest block for which (cid:80) x (cid:98) n x (cid:98) n x +1 = L/ N = (cid:0) L/ L/ (cid:1) ∼ L L , thus up topolynomial corrections in L , it still spans the full Hilbertspace of (cid:98) H , Eq. (G2). Nevertheless, even a further frag-mentation takes place in this block, which disjoints it intoexponential many blocks (cid:98) H ∞ (for details see Ref. [59]).We focus our analysis on the largest sub-block, whichhas the dimension n (cid:0) n/ − n/ (cid:1) ∼ √ L D Hilbert L , where D Hilbert = log − ≈ .
7. Thus, the eigenstates of (cid:98) H ∞ are confined in an exponentially small fraction ofthe full Hilbert space and their fractal dimensions, D q ,should be always smaller than unity D Hilbert <
1. Usingthe Wigner-Jordan transformations, one can show thatthe t − V is equivalent to the XXZ Heisenberg spin-1 / V → ∞ are multifractal as it has been shown forfinite anisotropy in XXZ model in [39].Now, we show numerically that even though the fractaldimensions of the eigenstates of (cid:98) H ∞ are strictly smallerthan one, the bipartite entanglement entropy S of thesestates can scale as the Page value ∼ L/ IP R andhalf-chain EE S for eigenstates in the middle of the spec-trum of (cid:98) H ∞ as a function of disorder strength W (seealso [59] for more details). IP R decreases exponentiallywith L , IP R ∼ − D L , N = 2 L . We can give an esti-mation of the fractal dimension D considering its finite-size approximation D ( L ) = − log IP R / log N . Up tofinite-size corrections, from the numerics we can deducethat 0 . ≤ D ≤ D ≤ D Hilbert ≈ . W ≤ .
25 asshown in Fig. 13 (c). At the same time, we can esti-mate the scaling of the EE D ent as D ent ( L ) = S / S P age ,see Fig. 13 (d). As one can see, D ent ≈ D ≤ ..