Higher-order spacing ratios in random matrix theory and complex quantum systems

Tekur, S. Harshini; Bhosale, Udaysinh T.; Santhanam, M. S.

doi:10.1103/physrevb.98.104305

Cited by 52 publications

(46 citation statements)

References 56 publications

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“…4 can be expected to be independent of T /N . Similar scaling relation had been postulated for the spacing distribution of Gaussian ensembles [34] and its generalizations [35], and recently numerical evidence was provided for the ratios [36]. Both P k (r, β) and P (r, β ′ ) have identical functional forms and the modified parameter β ′ depends on the order k of the spacing ratio and co-dimension β.…”

Section: Distribution Of Higher Order Spacing Ratiossupporting

confidence: 64%

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Scaling in the eigenvalue fluctuations of correlation matrices

2018

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The spectra of empirical correlation matrices, constructed from multivariate data, are widely used in many areas of sciences, engineering and social sciences as a tool to understand the information contained in typically large datasets. In the last two decades, random matrix theory-based tools such as the nearest neighbour eigenvalue spacing and eigenvector distributions have been employed to extract the significant modes of variability present in such empirical correlations. In this work, we present an alternative analysis in terms of the recently introduced spacing ratios, which does not require the cumbersome unfolding process. It is shown that the higher order spacing ratio distributions for the Wishart ensemble of random matrices, characterized by the Dyson index β, is related to the first order spacing ratio distribution with a modified value of co-dimension β ′ . This scaling is demonstrated for Wishart ensemble and also for the spectra of empirical correlation matrices drawn from the observed stock market and atmospheric pressure data. Using a combination of analytical and numerics, such scalings in spacing distributions are also discussed.

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Section: Distribution Of Higher Order Spacing Ratiossupporting

confidence: 64%

“…This finite size or convergence effect has also been discussed for Gaussian ensembles in Ref. [36]. While in the bottom panel, it is observed that the minima of D(β ′ ) does not converge to the predicted value β ′ = 229 even on considering a (relatively) small range of eigenvalues in the bulk.…”

Section: B Stock Market and Atmospheric Datasupporting

confidence: 60%

Scaling in the eigenvalue fluctuations of correlation matrices

2018

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“…Evidence for a broader class of interrelations involving β = 1, 2, 4 has been provided recently in Ref. [42]. Fig.…”

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confidence: 75%

Random Matrix Ensemble for the Level Statistics of Many-Body Localization

2019

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We numerically study the level statistics of the Gaussian β ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices β = 1, 2, 4 to the continuous range 0 < β < ∞. The Gaussian β ensemble covers Poissonian level statistics for β → 0, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian β ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.

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“…A straight-forward generalization, left for future work, would be to consider the statistics of the ratios s kCN i /s (k+1)CN i , which would give insight into longer range correlations. It is noteworthy that a recent work [55] explores certain higher range spacing ratios of a kind that is not based on ordering.…”

Section: Summary and Discussionmentioning

confidence: 99%

Ordered level spacing probability densities

Srivastava

Lakshminarayan

Tomsovic

et al. 2018

J. Phys. A: Math. Theor.

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Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a 3 × 3 matrix model. The closest neighbour density is generalized to the k-th closest neighbour spacing density, which allows for investigating long-range correlations. For larger k the probability density of k-th closest neighbour spacings is well described by a Gaussian. Using these k-th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found.

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Higher-order spacing ratios in random matrix theory and complex quantum systems

Cited by 52 publications

References 56 publications

Scaling in the eigenvalue fluctuations of correlation matrices

Scaling in the eigenvalue fluctuations of correlation matrices

Random Matrix Ensemble for the Level Statistics of Many-Body Localization

Ordered level spacing probability densities

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