Model of level statistics for disordered interacting quantum many-body systems

Abstract

We numerically study level statistics of disordered interacting quantum many-body systems. A two-parameter plasma model which controls level repulsion exponent β and range h of interactions between eigenvalues is shown to reproduce accurately features of level statistics across the transition from ergodic to many-body localized phase. Analysis of higher order spacing ratios indicates that the considered β-h model accounts even for long range spectral correlations and allows to obtain a clear picture of the flow of level statistics across the transition. Comparing spectral form factors of β-h model and of a system in the ergodic-MBL crossover, we show that the range of effective interactions between eigenvalues h is related to the Thouless time which marks the onset of quantum chaotic behavior of the system. Analysis of level statistics of random quantum circuit which hosts chaotic and localized phases supports the claim that β-h model grasps universal features of level statistics in transition between ergodic and many-body localized phases also for systems breaking time-reversal invariance.Many-body localization (MBL) [1, 2] is a robust phenomenon of ergodicity breaking in disordered interacting quantum many-body systems [3][4][5][6]. It has attracted considerable attention over the last decade, notable findings include an emergent integrability of MBL phase due to the existence of local integrals of motion (LIOMs) [3, 7-10] and an associated unbounded logarithmic growth of the bipartite entanglement entropy after a quench from a separable state [11,12]. A wide regime of subdiffusive transport on the ergodic side of the transition was found [13][14][15]. Signatures of MBL have been observed experimentally in 1D [16,17] and in 2D system [18], however, the stability of MBL in 2D is still debated [19].Spectral statistics of ergodic systems with (without) time reversal invariance follow predictions of Gaussian orthogonal (unitary) ensemble (GOE, GUE, respectively) of random matrices [20,21] while eigenvalues of localized systems are uncorrelated resulting in Poisson statistics (PS). A ratio of consecutive spacings between energy levelswas proposed as a simple probe of the level statistics in [22] with n = 1 and employed in investigation of ergodicity breaking in various settings [23][24][25][26][27][28][29][30]. Higher order spacing ratios (n > 1), studied in [31][32][33][34], are valuable tools to assess properties of level statistics. In contrast to standard measures such as level spacing distribution or number variance they do not require the so called unfolding, i.e. the procedure of setting density of energy levels ρ(E) to unity which can lead to misleading results [35].Recently, an analytical understanding of an appearance of random matrix theory statistics in systems without a clear semiclassical limit have been developed in a periodically driven Ising models [36,37] or in random Floquet circuits [38]. Variants of such systems have been argued to undergo ergodic-MBL transition [39,40].In this work we consider ...

Many-body localization transition in large quantum spin chains: The mobility edge

“…In this work, we consider open boundary conditions in the Hamiltonian (1). The random-field Heisenberg spin chain has been widely studied in the MBL context [12,31,[67][68][69][70][71][72][73], which has made it the de facto standard model of MBL studies.…”

Signatures of a critical point in the many-body localization transition

“…Finally, we wish to emphasize that other more common signatures of the transition from ergodicity to MBL, like the mean value of the adjacent level gap ratio or the family of Rényi entropies (of which the Shannon entropy is a particular case) [15,23,24,[36][37][38][39][40][41][42][43], change monotonically with ω, and therefore do not give rise to a neat singular point. For the previous indicators usually some form of scaling is involved in order to identify the transition point, and its value is generally largely influenced by several factors among which the most important is the number of simulated sites, L. By contrast, as can be seen in Figs.…”

“…Recent results indicate that it may be characterized by a Griffiths-like phase in which anomalously different disorder regions seem to dominate the dynamics [33][34][35]55]; nonetheless, the debate is still open [56]. To describe the flow of intermediate statistics observed in this region, mean-field plasma models with effective power-law interactions between energy levels [24,57], the Rosenzweig-Porter ensemble with multifractal eigenvectors [58,59], a family of short-range plasma models [60] and generalizations [42,43] have been used. Finally, for disorder strengths larger than a critical value that is dependent on the dimension of the Hilbert space, the chain gradually reaches the MBL phase.…”

“…Although exploring finite systems rather clearly leads to the second option, the answer in the thermodynamic limit (i.e., at macroscopic scales) is a lot more difficult to obtain, mainly due to the exponential increase of the dimensionality of the Hilbert space with system size in many-body quantum systems. A variety of ergodicity breaking indicators [15,23,24,28,[36][37][38][39][40][41][42][43] have been used in the past to try and identify a hypothetical value of disorder strength that, in the thermodynamic limit, completely separates the ergodic and the nonergodic phases. The search for such a critical point implicitly assumes a nature of real phase transition in MBL systems, although this has not been proved and some apparent contradictions seem to exist which have led to much debate in the community [28,[44][45][46][47].…”

Long-range level correlations in quantum systems with finite Hilbert space dimension

“…This kind of intermediate statistics is also present in nonrandom Hamiltonians with a steplike singularity [23], as well as in a Coulomb billiard [69], anisotropic Kepler problems [70], generalized kicked rotors [71], pseudointegrable billiards [22,24], and others [72]. Within the context of many-body quantum systems, several variations of this short-range plasma model and further generalizations have been put forward to describe the level statistics of the region between the ergodic (chaotic) and localized (integrable) phases in the many-body localization transition [35,56,73].…”