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

## TL;DR

This paper investigates the spectral properties of superposed random matrix ensembles using higher-order spacing ratios, deriving new relations and conjectures, with strong numerical evidence and applications to complex physical systems.

## Contribution

It introduces a novel analysis of superposed spectra via higher-order spacing ratios, establishing new scaling relations and conjectures for different ensembles and symmetry classes.

## Key findings

- Derived a modified index $eta'$ for superposed spectra.
- Established scaling relations between $eta'$ and $k$ for COE and CUE.
- Provided numerical evidence supporting the conjectures and applied results to physical systems.

## Abstract

The statistical properties of spectra of quantum systems within the framework of random matrix theory is widely used in many areas of physics. These properties are affected, if two or more sets of spectra are superposed, resulting from the discrete symmetries present in the system. Superposition of spectra of $m$ such circular orthogonal, unitary and symplectic ensembles are studied numerically using higher-order spacing ratios. For given $m$ and the Dyson index $\beta$, the modified index $\beta'$ is tabulated whose nearest neighbor spacing distribution is identical to that of $k$-th order spacing ratio. For the case of $m=2$ ($m=3$) in COE (CUE) a scaling relation between $\beta'$ and $k$ is given. For COE, it is conjectured that for $k=m+1$ ($m\geq2$) and $k=m-3$-th ($m\geq5$) order spacing ratio distribution the $\beta'$ is $m+2$ and $m-4$ respectively. Whereas in the case of CSE, for $k=m+1$ ($m\geq2$) and $k=m-1$-th ($m\geq3$) the $\beta'$ is $2m+3$ and $2(m-2)$ respectively. We also conjecture that for given $m$ ($k$) and $\beta$, the sequence of $\beta'$ as a function of $k$ ($m$) is unique. Strong numerical evidence in support of these results is presented. These results are tested on complex systems like the measured nuclear resonances, quantum chaotic kicked top and spin chains.