Random matrix model for eigenvalue statistics in random spin systems

Rao, Wen-Jia

doi:10.1016/j.physa.2021.126689

Cited by 7 publications

(2 citation statements)

References 40 publications

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“…In fact, from our study of uncorrelated eigenvalues, our results hold true for any β ≥ 0. These results have given analytical support to the numerical results from various random matrix ensembles and complex physical systems, which was absent earlier [9,34,[67][68][69]. Our analytical approach can be extended to other ensembles, for example Laguerre ensemble [2,3,56,[70][71][72], chiral ensemble [73][74][75][76][77][78][79], etc.…”

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confidence: 70%

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

Bhosale¹

2022

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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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