Long-range correlations in quantum-chaotic spectra

Pandey, Akhilesh; Puri, Sanjay; Kumar, Sandeep

doi:10.1103/physreve.71.066210

Cited by 12 publications

(8 citation statements)

References 25 publications

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“…Our results agree with those in [4] for τ = 0, ∞. We also remark that it should be possible to derive a long-range two-point correlation function analogous to those given in [24] for the transitions also. In fact such expansion has already been given for transitions in the Gaussian ensembles [2,14].…”

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confidence: 88%

Universal spectral correlations in orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices

Kumar¹,

Pandey²

2009

Phys. Rev. E

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Orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices are relevant in many contexts, especially in the study of time reversal symmetry breaking in quantum chaotic systems. Using skew-orthogonal polynomials we show that the same generic form of n-level correlation functions are obtained for the Jacobi family of crossover ensembles, including the Laguerre and Gaussian cases. For large matrices we find universal forms of unfolded correlation functions when expressed in terms of a rescaled transition parameter with arbitrary initial level density.

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confidence: 88%

Universal spectral correlations in orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices

Kumar¹,

Pandey²

2009

Phys. Rev. E

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“…Our MC approach follows [22] for the linear case of Eq. ( 1), and [23] for the circular case of Eq. ( 3).…”

Section: Monte Carlo Technique For Frcg Modelsmentioning

confidence: 99%

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

2020

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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...

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“…Similar generalizations can be done for circular ensembles, viz., ensembles of unitary matrices. The equilibrium properties of these non-Gaussian ensembles have been studied in detail [24,25]. Similarly non-uniform circular ensembles have also been studied [26].…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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 (FRCG) 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 new universality classes of random matrix ensembles. They also provide a theoretical framework to study banded random matrices, and dynamical systems whose matrix representation can be written in the form of banded matrices.

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Long-range correlations in quantum-chaotic spectra

Cited by 12 publications

References 25 publications

Universal spectral correlations in orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices

Universal spectral correlations in orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices

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

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

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