Scaling of the reduced energy spectrum of random matrix ensemble

Wen-Jia Rao; M. N. Chen

doi:10.1140/epjp/s13360-020-01067-3

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2024 Impact factor 2.9

EPJ PLUS - Condensed Matter and Complex Systems

Eur. Phys. J. Plus (2021) 136: 81
https://doi.org/10.1140/epjp/s13360-020-01067-3

Regular Article

Scaling of the reduced energy spectrum of random matrix ensemble

Wen-Jia Rao^a and M. N. Chen

School of Science, Hangzhou Dianzi University, 310027, Hangzhou, China

^a wjrao@hdu.edu.cn

Received: 8 October 2020
Accepted: 31 December 2020
Published online: 13 January 2021

Abstract

We study the reduced energy spectrum ${E_{i}^{(n)}}$ $\{E_{i}^{(n)}\}$ , which is constructed by picking one level from every n levels of the original spectrum ${E_{i}}$ $\{E_{i}\}$ , in a Gaussian ensemble of random matrix with Dyson index $β \in (0, \infty)$ $\beta \in \left( 0,\infty \right)$ . It is shown the joint probability distribution of ${E_{i}^{(n)}}$ $\{E_{i}^{(n)}\}$ bears the same form as ${E_{i}}$ $\{E_{i}\}$ with a rescaled parameter $γ = \frac{n (n + 1)}{2} β + n - 1$ $\gamma =\frac{n(n+1)}{2}\beta +n-1$ . Notably, the nth-order level spacing and gap ratio in ${E_{i}}$ $\{E_{i}\}$ become the lowest order ones in ${E_{i}^{(n)}}$ $\{E_{i}^{(n)}\}$ , which explains their distributions found separately by recent studies in a consistent way. Our results also establish the higher-order spacing distributions in random matrix ensembles beyond GOE, GUE and GSE and reveal a hierarchy of structures hidden in the energy spectrum.