Improved unfolding by detrending of statistical fluctuations in quantum spectra

Morales, Irving O.; Landa, Emmanuel; Stránský, Pavel; Frank, A.

doi:10.1103/physreve.84.016203

Cited by 28 publications

(24 citation statements)

References 21 publications

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“…In a recent approach, the unfolded fluctuations of the accumulated level density function N (E) = N (E) − N (E) (also called δ n function) * Email: fossion@nucleares.unam.mx were interpreted as a time series [4,20,21]. This treatment opened the field to the application of specialized techniques from signal analysis, such as Fourier spectral analysis [4,7,20,21], Detrended Fluctuation Analysis (DFA) [22][23][24], wavelets [25], Empirical Mode Decomposition (EMD) [26][27][28], and normal-mode analysis [29,30]. The result of these investigations is that for Gaussian RMT ensembles, the fluctuation time series is scale invariant (fractal), which in the Fourier power spectrum is reflected in a power law,…”

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confidence: 99%

Random-matrix spectra as a time series

2013

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Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a time series. Dataadaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding possible artifacts introduced by standard unfolding techniques. The fluctuation modes are scale invariant and follow different power laws for Poisson and Gaussian ensembles, which already during the unfolding allows to distinguish the two cases.PACS numbers: 05.45. Tp,05.45.Mt,02.50.Sk The study of spectral fluctuations within the framework of Random Matrix Theory (RMT) is a standard tool in the statistical study of quantum chaos in the excitation spectra of quantum systems [1][2][3][4]. Recently, the approach has found new applications in many fields, such as in the study of eigenspectra of adjacency matrices of networks [5][6][7], and eigenspectra of empirical correlation matrices in finance [8][9][10], the climate [11], electro-and magnetoencephalography [12][13][14], and in complex systems [15]. The interest of the approach lies in the fact that the level density fluctuations ρ(E) = ρ(E) − ρ(E) around the smooth global density ρ(E) are universal and indicate the underlying symmetry class of the system [2,16]. On the other hand, the global level density ρ(E) is system dependent, and an unfolding procedure needs to be performed, to separate the global and the fluctuating parts [1]. The unfolding is straightforward if an analytical formula is known to describe the global level density ρ(E) for the system under study, such as e.g. the gaussian and semicircle distributions for Possion and GOE matrix ensembles from RMT [2], or the MarchenkoPastur distribution for the Laguerre ensemble of random Wishart correlation matrices [17]. However, such analytical formulae are formally only adequate in the asymptotic limit for spectra with an infinite number of levels. Often, an analytical form for ρ(E) is unknown, as is the case for adjacency matrices [5]. In practical cases, having finite, albeit large matrices, the usual approach is then to project the sequence of ordered eigenvalues into unfolded values E(n) → N [E(n)], using a smooth (often polynomial) approximation N (E) to the accumulated density , 5, 20]. After unfolding, the short-range and long-range correlations can be quantified using standard fluctuations measures such as the Nearest-Neighbour Spacing (NNS) distribution, number variance Σ 2 and ∆ 3 . In a recent approach, the unfolded fluctuations of the accumulated level density function N (E) = N (E) − N (E) (also called δ n function) *

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confidence: 99%

Random-matrix spectra as a time series

2013

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“…The distinction between these two parts of ρ(E) is to some extent arbitrary and may lead to dubious outcomes see, for instance, the differences between local unfolding and Gaussian broadening [35]. Over the past years, there is a constant effort to tackle such problems from the mathematical and computational viewpoints [36,37].…”

Section: The Interpolating Ensembles and Formulaementioning

confidence: 99%

On intermediate statistics across many-body localization transition

De¹,

Sierant²,

Zakrzewski³

2021

J. Phys. A: Math. Theor.

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The level statistics in the transition between delocalized and localized {phases of} many body interacting systems is {considered}. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically {via Monte Carlo method}. The resulting higher order spacing ratios are compared with data coming from different {quantum many body systems}. It is found that this Pechukas-Yukawa distribution compares favorably with {$\beta$--Gaussian ensemble -- a single parameter model of level statistics proposed recently in the context of disordered many-body systems.} {Moreover, the Pechukas-Yukawa distribution is also} only slightly inferior to the two-parameter $\beta$-h ansatz shown {earlier} to reproduce {level statistics of} physical systems remarkably well.

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“…The distinction between these two parts of ρ(E) is to some extent arbitrary and may lead to dubious outcomes see, for instance, the differences between local unfolding and Gaussian broadening [33]. Over the past years, there is a constant effort to tackle such problems from the mathematical and computational viewpoints [34,35].…”

Section: The Interpolating Ensembles and Formulaementioning

confidence: 99%

On intermediate statistics across many-body localization transition

De,

Sierant,

Zakrzewski

2021

Preprint

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The level statistics in the transition between delocalized and localized phases of many body interacting systems is considered. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically via Monte Carlo method. The resulting higher order spacing ratios are compared with data coming from different quantum many body systems. It is found that this Pechukas-Yukawa distribution compares favorably with β-Gaussian ensemble -a single parameter model of level statistics proposed recently in the context of disordered many-body systems. Moreover, the Pechukas-Yukawa distribution is also only slightly inferior to the two-parameter β-h ansatz shown earlier to reproduce level statistics of physical systems remarkably well.

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Improved unfolding by detrending of statistical fluctuations in quantum spectra

Cited by 28 publications

References 21 publications

Random-matrix spectra as a time series

Random-matrix spectra as a time series

On intermediate statistics across many-body localization transition

On intermediate statistics across many-body localization transition

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