We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures-Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Havrda-Charvát-Tsallis (HCT) entropy as a finite sum. Averages of von Neumann entropy, linear entropy and purity follow by considering appropriate limits in the average HCT expression. Based on exact evaluations of the average von Neumann entropy and the average purity, we also conjecture very simple formulae for these, which are similar to those in the Hilbert-Schmidt ensemble.

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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We systematically study the short range spectral fluctuation properties of three non-hermitian spin chain hamiltonians using complex spacing ratios. In particular we focus on the non-hermitian version of the standard one-dimensional anisotropic XY model having intrinsic rotation-time-reversal (RT ) symmetry that has been explored analytically by Zhang and Song in [Phys.Rev.A 87, 012114 (2013)]. The corresponding hermitian counterpart is also exactly solvable and has been widely employed as a toy model in several condensed matter physics problems. We show that the presence of a random field along the x-direction together with the one along z facilitates integrability and RT -symmetry breaking leading to the emergence of quantum chaotic behaviour indicated by a spectral crossover resembling Poissonian to Ginibre unitary ensemble (GinUE) statistics of random matrix theory. Additionally, we consider two n × n dimensional phenomenological random matrix models in which, depending upon crossover parameters, the fluctuation properties measured by the complex spacing ratios show an interpolation between 1D-Poisson to GinUE and 2D-Poisson to GinUE behaviour. Here 1D and 2D Poisson correspond to real and complex uncorrelated levels, respectively.