Finite-range Coulomb gas models of banded random matrices and quantum kicked rotors

Enhancement in breaking of time-reversal invariance in the quantum kicked rotor

We study the breaking of time-reversal invariance (TRI) by the application of a magnetic field in the quantum kicked rotor (QKR), using Izrailev's finite-dimensional model. There is a continuous crossover from TRI to time-reversal noninvariance (TRNI) in the spectral and eigenvector fluctuations of the QKR. We show that the properties of this TRI to TRNI transition depend on α^{2}/N, where α is the chaos parameter of the QKR and N is the dimensionality of the evolution operator matrix. For α^{2}/N≳N, the transition coincides with that in random matrix theory. For α^{2}/N<N, the transition shows a marked deviation from random matrix theory. Further, the speed of this transition as a function of the magnetic field is significantly enhanced as α^{2}/N decreases.

Particles confined in arbitrary potentials with a class of finite-range repulsive interactions

In this paper, we develop a large N field theory for a system of N classical particles in one dimension at thermal equilibrium. The particles are confined by an arbitrary external potential, V_{ex}(x), and repel each other via a class of pairwise interaction potentials V_{int}(r) (where r is distance between a pair of particles) such that V_{int}∼|r|^{-k} when r→0. We consider the case where every particle is interacting with d (finite-range parameter) number of particles to its left and right. Due to the intricate interplay between external confinement, pairwise repulsion, and entropy, the density exhibits markedly distinct behavior in three regimes k>0, k→0, and k<0. From this field theory, we compute analytically the average density profile for large N in these regimes. We show that the contribution from interaction dominates the collective behavior for k>0 and the entropy contribution dominates for k<0, and both contribute equivalently in the k→0 limit (finite-range log-gas). Given the fact that this family of systems is of broad relevance, our analytical findings are of paramount importance. These results are in excellent agreement with brute-force Monte Carlo simulations.

Finite-range Coulomb gas models. I. Some analytical results

Dyson has shown an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. In this paper, we introduce finite-range Coulomb gas models as a generalization of the Dyson models with a finite range of eigenvalue interactions. As the range of interaction increases, there is a transition from Poisson statistics to classical random matrix statistics. These models yield distinct universality classes of random matrix ensembles. They also provide a theoretical framework to study banded random matrices, and dynamical systems the matrix representation of which can be written in the form of banded matrices.

Finite-range Coulomb gas models. II. Applications to quantum kicked rotors and banded random matrices

In part I of this two-stage exposition [Pandey, Kumar, and Puri, preceding paper, Phys. Rev. E 101, 022217 (2020)10.1103/PhysRevE.101.022217], we introduced finite-range Coulomb gas (FRCG) models, and developed an integral-equation framework for their study. We obtained exact analytical results for d=0,1,2, where d denotes the range of eigenvalue interaction. We found that the integral-equation framework was not analytically tractable for higher values of d. In this paper, we develop a Monte Carlo (MC) technique to study FRCG models. Our MC simulations provide a solution of FRCG models for arbitrary d. We show that, as d increases, there is a transition from Poisson to Wigner-Dyson classical random matrix statistics. Thus FRCG models provide a route for transition from Poisson to Wigner-Dyson statistics. The analytical formulation obtained in part I, and MC techniques developed in this paper, are used to study banded random matrices (BRMs) and quantum kicked rotors (QKRs). We demonstrate that, for a BRM of bandwidth b and a QKR of chaos parameter α, the appropriate FRCG model has range d=b^{2}/N=α^{2}/N, for N→∞. Here, N is the dimensionality of the matrix in the BRM, and the evolution operator matrix in the QKR.

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.