Higher-order spacings in the superposed spectra of random matrices with comparison to spacing ratios and application to complex systems

TL;DR

The paper develops and tests higher-order spacing statistics for the $m$-fold superposition of circular random matrices (COE, CUE, CSE) to infer the symmetry structure of complex quantum systems without desymmetrizing spectra. By fitting the observed $k$-th order spacings to a nearest-neighbor model with a modified Dyson index $β'$, the authors tabulate $β'$ as a function of $m$, $k$, and $β$, and hypothesize a unique $β'(k)$ sequence for fixed $(m,β)$. They verify scaling relations and apply the framework to physical models such as the intermediate map and the quantum kicked top, revealing overall agreement with RMT predictions up to moderate $k$ and highlighting finite-size, randomness, and seed dependencies at larger $k$. The work provides a practical, unfolding-free tool for symmetry detection and fluctuation characterization in complex systems, with potential applications in quantum chaos and many-body physics, and points to avenues for full analytical treatments of HOS and HOSR in the bulk of circular ensembles.

Abstract

The connection between random matrices and the spectral fluctuations of complex quantum systems in a suitable limit can be explained by using the setup of random matrix theory. Higher-order spacing statistics in the $m$ superposed spectra of circular random matrices are studied numerically. We tabulated the modified Dyson index $β'$ for a given $m$, $k$, and $β$, for which the nearest neighbor spacing distribution is the same as that of the $k$-th order spacing distribution corresponding to the $β$ and $m$. Here, we conjecture that for given $m(k)$ and $β$, the obtained sequence of $β'$ as a function of $k(m)$ is unique. This result can be used as a tool for the characterization of the system and to determine the symmetry structure of the system without desymmetrization of the spectra. We verify the results of the $m=2$ case of COE with the quantum kicked top model corresponding to various Hilbert space dimensions. From the comparative study of the higher-order spacings and ratios in both $m=1$ and $m=2$ cases of COE and GOE by varying dimension, keeping the number of realizations constant and vice-versa, we find that both COE and GOE have the same asymptotic behavior in terms of a given higher-order statistics. But, we found from our numerical study that within a given ensemble of COE or GOE, the results of spacings and ratios agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. It is observed that for the $k=1$ case, the convergence towards the Poisson distribution is faster in the case of ratios than the corresponding spacings as we increase $m$ for a given $β$. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. There, we find that the effect of random numbers used to generate the matrix corresponding to the map is reflected in the higher-order statistics.

Figures (59)

• Figure 1: Distribution of the $k$-th order spacings (circles) for the superposition of $m=2$ COE spectra. Here, $N=5000$ and $n=600$. The solid curve corresponds to $P(s,\beta')$ as given in Eq. (\ref{['Eq:PSBeta']}), in which $\beta$ is replaced by $\beta'$ and $\beta'$ is given in Table \ref{['Table:COEtable1']}. The insets show $D(\beta')$ as a function of $\beta'$.
• Figure 2: Same as Fig. \ref{['fig:COE2added']} but for different values of $k$ and $\beta'$.
• Figure 3: Same as Fig. \ref{['fig:COE2added']} but for $m=5$, $n=1000$, and different values of $k$ and $\beta'$.
• Figure 4: Same as Fig. \ref{['fig:COE2added']} but for $m=6$, $n=1002$, and different values of $k$ and $\beta'$.
• Figure 5: Same as Fig. \ref{['fig:COE2added']} but for $m=7$, $n=1001$, and different values of $k$ and $\beta'$.