Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles
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The ratio of two consecutive level spacings has emerged as a very useful metric in exploring universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in studying 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 result for the distribution of ratio of two consecutive level spacings in the Gaussian orthogonal to unitary crossover ensemble using a 3× 3 random matrix model. This crossover is useful in modeling time reversal symmetry breaking in quantum chaotic systems. Although based on a 3× 3 matrix model, our result can be applied in the study of large spectra also, 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.
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