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

The statistical properties of spectra of quantum systems within the framework of random matrix theory are 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. The 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 $\ensuremath{\beta}$, the modified index ${\ensuremath{\beta}}^{\ensuremath{'}}$ 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 the Circular Orthogonal Ensemble (Circular Unitary Ensemble) a scaling relation between ${\ensuremath{\beta}}^{\ensuremath{'}}$ and $k$ is given. For COE, it is conjectured that for $k=m+1$ ($m\ensuremath{\ge}2$) and $k=m\ensuremath{-}3$-th ($m\ensuremath{\ge}5$) order spacing ratio distribution the ${\ensuremath{\beta}}^{\ensuremath{'}}$ is $m+2$ and $m\ensuremath{-}4$, respectively, whereas in the case of the Circular Symplectic Ensemble, for $k=m+1$ ($m\ensuremath{\ge}2$) and $k=m\ensuremath{-}1$-th ($m\ensuremath{\ge}3$) the ${\ensuremath{\beta}}^{\ensuremath{'}}$ is $2m+3$ and $2(m\ensuremath{-}2)$, respectively. We also conjecture that for given $m$ ($k$) and $\ensuremath{\beta}$, the sequence of ${\ensuremath{\beta}}^{\ensuremath{'}}$ 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.