Superposition and higher-order spacing ratios in random matrix theory with application to complex systems

Bhosale, Udaysinh T.

doi:10.1103/physrevb.104.054204

Cited by 6 publications

(3 citation statements)

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“…This relation has been employed successfully to various physical systems like chaotic billiards, Floquet systems, circular ensembles, spin chains, observed stock market, etc. [9,28,[55][56][57][58][59] , to estimate the number of symmetries in complex physical systems [44,45]. It should be noted that, a similar scaling relation between the higherorder and NN spacing distributions has be proposed earlier in Refs.…”

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confidence: 84%

“…where Z β is the normalization constant. This distribution has found many applications, to study eigenvalue statistics in spin systems [25,[28][29][30][31][32][33][34][35], in triangular billiards [36], in the Hessians of artificial neural networks [37], in Sachdev-Ye-Kitaev model [38][39][40][41][42], in quantum field theory [43], to quantify symmetries in various complex systems [44,45].…”

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confidence: 99%

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Universal scaling of higher-order spacing ratios in Gaussian random matrices

Bhosale¹

2022

Preprint

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Higher-order spacing ratios in Gaussian ensembles are investigated analytically. A universal scaling relation, known from earlier numerical studies, of the higher-order spacing ratios is proved in the asymptotic limits. I. INTRODUCTIONRandom matrix theory (RMT), introduced for more than fifty years, has been applied successfully in various fields [1][2][3]. Originally it was introduced to explain complex spectra of heavy nucleus [4]. Later, it has found applications in complex networks [5,6], many-body physics [7][8][9][10][11], wireless communications [12], etc. One of the main objective of RMT is to study the spectral fluctuations in these systems. These fluctuations can be used to characterize the different types phases of these complex systems. For example, integrable to chaotic limits of the underlying classical systems [13][14][15], thermal or localized phases of condensed matter systems [9-11, 16], etc. Bohigas, Giannoni, and Schmit conjectured that the eigenvalue fluctuations in a quantum chaotic system can be modelled by any one of the three classical ensembles of RMT. These ensembles having Dyson indices as β = 1, 2 and 3 respectively corresponds to Hermitian random matrices whose entries are chosen/distributed independently, respectively, as real (GOE), complex (GUE), or quaternionic (GSE) random variables [1].The most popular measure to model the spectral fluctuations is the nearest neighbour (NN) level spacings, s i = E i+1 − E i , where E i , i = 1, 2, . . . are the eigenvalues of the given Hamiltonian H. A surmise by Wigner states that in a time-reversal invariant system (β = 1) which do not have a spin degree of freedom, these spacings are distributed as P (s) = (π/2)s exp(−πs 2 /4), which indicates the level repulsion. This result is very close the exact one which has been obtained later on [1,3,17]. For such systems, Gaussian Orthogonal Ensemble (GOE) is well suited to study the statistical properties of their spectra. There are other ensembles also commonly used in RMT, namely, Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE) having Dyson index β = 2 and 4 respectively. These ensembles have been implemented successfully in various fields [2,18]. In this paper, the Gaussian ensembles are studied in detail and various analytical results are obtained.

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Universal scaling of higher-order spacing ratios in Gaussian random matrices

Bhosale¹

2022

Preprint

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“…In addition, our results indicate an intriguing correlation between singlets and doublets spectra for the E-C 3 B family, producing p(s)'s that move away from the GOE and GUE distributions as ρ c decreases, thus requiring a further investigation of the ob-served effect. In this perspective, a range opens up to investigate the correlation of spectra of different subspaces [26,34,[46][47][48] in billiards that present only rotational symmetries greater than three, which will give the possibility of performing other tests with the new formulas (8) and (10).…”

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Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets

Araújo¹

2023

Preprint

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Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory these systems are simple prototypes, their conservative dynamics of a billiard may vary from regular to chaotic, depending only on the border. The results reported here seek to shed light on the quantization of classically chaotic systems. We present numerical results on classical and quantum properties in two bi-parametric families of Billiards, Elliptical Stadium Billiard (ESB) and Elliptical-C3 Billiards (E-C3B). Both are elliptical perturbations of chaotic billiards with originally circular sectors on their borders. Our numerical calculations show evidence that the elliptical families can present a mixed classical phase space, identified by a parameter ρc < 1, which we use to guide our analysis of quantum spectra. We explored the short correlations through nearest neighbor spacing distribution p(s), which showed that in the mixed region of the classical phase space, p(s) is well described by the Berry-Robnik-Brody (BRB) distributions for the ESB. In agreement with the expected from the so-called ergodic parameter α = tH/tT, the ratio between the Heisenberg time and the classical diffusive-like transport time signals the possibility of quantum dynamical localization when α < 1. For the E-C3B family, the eigenstates can be split into singlets and doublets. BRB describes p(s) for singlets as the previous family in the mixed region. However, the p(s) for doublets are described by new distributions recently introduced in the literature but only tested in a few cases for ρc < 1. We observed that as ρc decreases, the p(s)'s tend to move away simultaneously from the GOE (singlets) and GUE (doublets) distributions.

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Signatures of quantum integrability and exactly solvable dynamics in an infinite-range many-body Floquet spin system

Sharma,

Bhosale

2024

Phys. Rev. B

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Superposition and higher-order spacing ratios in random matrix theory with application to complex systems

Cited by 6 publications

References 112 publications

Universal scaling of higher-order spacing ratios in Gaussian random matrices

Universal scaling of higher-order spacing ratios in Gaussian random matrices

Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets

Signatures of quantum integrability and exactly solvable dynamics in an infinite-range many-body Floquet spin system

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