Joint probability densities of level spacing ratios in random matrices

Atas, Y. Y.; Bogomolny, E.; Giraud, Olivier; Vivo, Pierpaolo; Vivo, Edoardo

doi:10.1088/1751-8113/46/35/355204

Cited by 103 publications

(84 citation statements)

References 38 publications

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“…Interestingly, we find that (S.3) is fulfilled with an excellent agreement for β-h model as our Monte Carlo simulation shows for arbitrary real β ∈ [0, 2] and h ∈ [1,40]. A number of analytical results is available for h = 1 [82,83]. A straightforward application of the method of [41] shows that distributions of higher order spacing ratios for h = 1 are given by P (r (n) ) = (r (n) ) β+(β+1)(n−1) ((β + 1)((r (n) ) + 1)) 2(β+1)n .…”

Section: Analytical Expressions For β-H Modelsupporting

confidence: 71%

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

Sierant

Zakrzewski

2020

Phys. Rev. B

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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 ...

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Section: Analytical Expressions For β-H Modelsupporting

confidence: 71%

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

Sierant

Zakrzewski

2020

Phys. Rev. B

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“…Those has been successfully used in the transition between MBL and extended phases [24,29]. The corresponding P (r) may be analytically determined [28] and is given by:…”

Section: Small System Sizes -Level Statistics Approachmentioning

confidence: 99%

“…While the inset shows the limiting cases of GOE and Poisson distributions, the intermediate statistics in the transition regime is intricate. Close to the localized side (U ≥ 10), one can use again the generalized semi-Poisson distribution (see above) whose prediction is [28] …”

Section: Small System Sizes -Level Statistics Approachmentioning

confidence: 99%

Many-Body Localization for Randomly Interacting Bosons

2017

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We study many-body localization in a one dimensional optical lattice filled with bosons. The interaction between bosons is assumed to be random, which can be realized for atoms close to a microchip exposed to a spatially fluctuating magnetic field. Close to a Feshbach resonance, such controlled fluctuations can be transfered to the interaction strength. We show that the system reveals an inverted mobility edge, with mobile particles at the lower edge of the spectrum. A statistical analysis of level spacings allows us to characterize the transition between localized and excited states. The existence of the mobility edge is confirmed in large systems, by time dependent numerical simulations using tDMRG. A simple analytical model predicts the long time behavior of the system.

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“…In the limits r → 0 and r → ∞, the asymptotic behaviors of these functions are r β and r −β−2 , respectively, which coincide with the corresponding asymptotic behaviors of GOE and GUE results in Eqs. (16) and (18). However, overall shapes of these distributions based on 3×3 cases of Laguerre and Gaussian ensembles are very distinct.…”

Section: Application To Laguerre Ensemblementioning

confidence: 95%

Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles

2020

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The ratio of two consecutive level spacings has emerged as a very useful metric in exploring universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in studying the spectrum of a general system. The Wigner surmise like results for the ratio distribution are known for the invariant classes of Gaussian random matrices. However, for the crossover ensembles, which are useful in modeling systems with partially broken symmetries, corresponding results have remained unavailable so far. In this work, we derive exact result for the distribution of ratio of two consecutive level spacings in the Gaussian orthogonal to unitary crossover ensemble using a 3 × 3 random matrix model. This crossover is useful in modeling time reversal symmetry breaking in quantum chaotic systems. Although based on a 3 × 3 matrix model, our result can be applied in the study of large spectra also, provided the symmetry breaking parameter facilitating the crossover is suitably scaled. We substantiate this claim by considering Gaussian and Laguerre crossover ensembles comprising large matrices. Moreover, we apply our result to investigate the violation of time-reversal invariance in the quantum kicked rotor system. *

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Joint probability densities of level spacing ratios in random matrices

Cited by 103 publications

References 38 publications

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

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

Many-Body Localization for Randomly Interacting Bosons

Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles

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