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

Abstract

In paper I of this two-stage exposition, 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 II, 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 novel route for transition from Poisson to Wigner-Dyson statistics. The analytical formulation obtained in paper I, and MC techniques developed in this paper II, are used to study banded random matrices (BRM) and quantum kicked rotors (QKR). 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 BRM, and the evolution operator matrix in QKR.Random matrices [1-4] have found extensive applications in quantum chaos, i.e., the study of quantum systems whose classical counterpart are chaotic [5][6][7][8][9][10]. The connection between quantum chaos and random matrices is well established. An important paradigm of quantum chaotic systems is the quantum kicked rotor (QKR) [7,8,10]. The Hamiltonian of the QKR is periodic in time with a delta-function perturbation. In paper I, we introduced and analytically studied finite-range Coulomb gas (FRCG) models which define novel classes of random-matrix ensembles. These are parametrized by the range of eigenvalue interactions, denoted as d.In this paper II, we demonstrate the applicability of FRCG models to quantum chaotic systems. We show that spectral fluctuations of the time evolution operator over one period (i.e., the Floquet operator) of the QKR can be modeled by FRCG models. In other applications, FRCG models have also been used to study quantum pseudo-integrable systems [11,12]. We expect that they would also be applicable to many other physical systems.An unusual property of quantum chaos is the suppression of chaotic diffusion. In classically chaotic systems such as the classical kicked rotor, the average energy of the system grows linearly with time. However, in its quantum chaotic counterpart (i.e., the QKR), the average energy saturates in time. The suppression of diffusion in the QKR is also known as dynamical localization [13]. The origin of this phenomenon lies in the localization of wave-functions of the Floquet operator in the momentum basis [14][15][16][17]. In this context, many studies have focused on the transition from ergodic (Wigner-Dyson statistics) to integrable behavior (Poisson statistics) in disordered systems [16][17][18][19]. The operator corresponding to the Hamiltonian of quantum systems exhibiting localization can be described by banded random matrices (BRM). T...

Enhancement in Breaking of Time-reversal Invariance in the Quantum Kicked Rotor

“…In Dyson's model for circular ensembles, the perturbation V (τ ), when represented in U(τ )-diagonal basis, belongs to Gaussian ensembles of random matrices. On the other hand, in FRCG models, V (τ ) takes the form of band matrices [26][27][28]. The Brownian motion of matrix elements in Eq.…”

“…We report results from a detailed numerical study of this transition. The motivation for this work comes from our recent proposal of finite-range Coulomb gas (FRCG) models for random matrix ensembles, and their applicability to a finite-dimensional matrix model of the QKR [26][27][28]. The CKR is characterized by a parameter α, with the system being highly chaotic for α ≫ 1.…”

“…The relevant parameter in the QKR is α 2 /N, where N is the size of the evolution operator matrix [26,29]. In our recent work [26][27][28], we have proposed that the spectral statistics of the QKR with chaos parameter α is described by an FRCG model of range d = α 2 /N. These FRCG models are generalizations of the Dyson's Coulomb gas model to the case where eigenvalues have a restricted interaction range.…”