# Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles

## TL;DR

This paper derives exact formulas for the distribution of the ratio of consecutive level spacings in Gaussian crossover ensembles, aiding the analysis of symmetry-breaking effects in quantum chaotic systems.

## Contribution

It provides the first exact results for the ratio distribution in Gaussian orthogonal to unitary crossover ensembles using a 3x3 matrix model, applicable to large spectra.

## Key findings

- Derived exact ratio distribution formulas for crossover ensembles.
- Validated results with large matrix simulations.
- Applied findings to quantum kicked rotor system.

## Abstract

The ratio of two consecutive level spacings has emerged as a very useful metric in investigating universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in analyzing the spectrum of a general system. The Wigner-surmise-like results for the ratio distribution are known for the invariant classes of Gaussian random matrices. However, for the crossover ensembles, which are useful in modeling systems with partially broken symmetries, corresponding results have remained unavailable so far. In this work, we derive exact results for the distribution and average of the ratio of two consecutive level spacings in the Gaussian orthogonal to unitary crossover ensemble using a $3\times 3$ random matrix model. This crossover is useful in modeling time-reversal symmetry breaking in quantum chaotic systems. Although based on a $3\times 3$ matrix model, our results can also be applied in the study of large spectra, provided the symmetry-breaking parameter facilitating the crossover is suitably scaled. We substantiate this claim by considering Gaussian and Laguerre crossover ensembles comprising large matrices. Moreover, we apply our result to investigate the violation of time-reversal invariance in the quantum kicked rotor system.