Multifractal dimensions for orthogonal-to-unitary crossover ensemble.

Multifractal analysis is a powerful approach for characterizing ergodic or localized nature of eigenstates in complex quantum systems. In this context, the eigenvectors of random matrices belonging to invariant ensembles naturally serve as models for ergodic states. However, it has been found that the finite-size versions of multifractal dimensions for these eigenvectors converge to unity logarithmically slowly with increasing system size N. In fact, this strong finite-size effect is capable of distinguishing the ergodicity behavior of orthogonal and unitary invariant classes. Motivated by this observation, in this work, we provide semi-analytical expressions for the ensemble-averaged multifractal dimensions associated with eigenvectors in the orthogonal-to-unitary crossover ensemble. Additionally, we explore shifted and scaled variants of multifractal dimensions, which, in contrast to the multifractal dimensions themselves, yield distinct values in the orthogonal and unitary limits as N→∞ and, therefore, may serve as a convenient measure for studying the crossover. We substantiate our results using Monte Carlo simulations of the underlying crossover random matrix model. We then apply our results to analyze the multifractal dimensions in a quantum kicked rotor, a Sinai billiard system, and a correlated spin-chain model in a random field. The orthogonal-to-unitary crossover in these systems is realized by tuning relevant system parameters, and we find that in the crossover regime, the observed finite-dimension multifractal dimensions can be captured very well with our results.

Spectral crossovers in non-Hermitian spin chains: Comparison with random matrix theory.

We present a systematic investigation of the short-range spectral fluctuation properties of three non-Hermitian spin-chain Hamiltonians using complex spacing ratios (CSRs). Specifically, we focus on the non-Hermitian variants of the standard one-dimensional anisotropic XY model having intrinsic rotation-time (RT) symmetry that has been explored analytically by Zhang and Song [Phys. Rev. A 87, 012114 (2013)1050-294710.1103/PhysRevA.87.012114]. 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 the z direction facilitates integrability and RT-symmetry breaking, leading to the emergence of quantum chaotic behavior. This is evidenced by a spectral crossover closely resembling the transition from Poissonian to Ginibre unitary ensemble (GinUE) statistics of random matrix theory. Additionally, we consider two phenomenological random matrix models in this paper to examine 1D Poisson to GinUE and 2D Poisson to GinUE crossovers and the associated signatures in CSRs. Here 1D and 2D Poisson correspond to real and complex uncorrelated levels, respectively. These crossovers reasonably capture spectral fluctuations observed in the spin-chain systems within a certain range of parameters.

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

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 3×3 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 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.