Scaling of the Reduced Energy Spectrum of Random Matrix Ensemble

Abstract

We study the reduced energy spectrum $\{E_i^{(n)}\}$, which is constructed by picking one level from every $n$ levels of the original spectrum $\{E_i\}$, in a Gaussian ensemble of random matrix with Dyson index $\beta \in \left(0,\infty \right)$. It's shown $\{E_i^{(n)}\}$ bears the same form of probability distribution as $\{E_i\}$ with a rescaled parameter $\gamma =\frac{n(n+1)}{2}\beta +n-1$. Notably, the $n$-th order level spacing and non-overlapping gap ratio in $\{E_i\}$ become the lowest-order ones in $\{E_i^{(n)}\}$, hence their distributions will rescale in an identical way. Numerical evidences are provided by simulating random spin chain as well as modelling random matrices. Our results establish the higher-order spacing distributions in random matrix ensembles beyond GOE,GUE,GSE, and reveals a hierarchy of structures hidden in the energy spectrum.

8 pages, 4 figures

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Cited by in corpus (2)

• Critical Level Statistics at the Many-Body Localization Transition Region
• Random Matrix Model for Eigenvalue Statistics in Random Spin Systems