Abstract
This is a collection of notes that are about spectral form factors of standard ensembles in the random matrix theory, written for the practical usage of current study of late time quantum chaos. More precisely, we consider Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal Ensemble (GOE), Gaussian Symplectic Ensemble (GSE), Wishart-Laguerre Unitary Ensemble (LUE), Wishart-Laguerre Orthogonal Ensemble (LOE), and Wishart-Laguerre Symplectic Ensemble (LSE). These results and their physics applications cover a three-fold classification of late time quantum chaos in terms of spectral form factors.
Highlights
This is a collection of notes about spectral form factors of standard ensembles in random matrix theory, written for the practical usage of the current study of late time quantum chaos
Following the pioneering works done by Wigner [1] and Dyson [2], people regard random matrix theory as a tool to classify a generic random Hamiltonian with discrete symmetries, and their energy spectra have been observed to satisfy universal behaviors [3,4,5]
The scientific interest of random matrix theory varies from nonlinear science, mathematics, and mathematical physics, to nuclear physics, statistical physics, and quantum field theory. (See, for instance, [6,7,8,9,10] for reference.) Some recent discoveries about black hole physics have lead to interest in the understanding of scrambling properties of quantum chaotic systems [11,12,13], where people start to consider an early chaotic behavior
Summary
The theory of quantum chaos, and its connection to random matrix theory, have several new recent developments on understanding novel behaviors of condensed matter systems and the quantum nature of black hole physics. Some further investigations show that the spectral form factor is one of the key roles serving in several quantum chaotic systems, and it could connect to out-of-time-ordered correlators and some other chaotic diagnostics [23,24,25]. Those facts motivate us to study the spectral form factor in random matrix theory and its mathematical properties, in detail, from a modern chaotic physicist point of view.
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