Two-spin models with classical chaos and different quantum universality classes.

Abstract

The local fluctuations in the quantum energy spectrum of a classically completely chaotic autonomous dynamical system are expected to be the same as those in the eigenvalues of Gaussian random Hermitian matrices. That relationship between a dynamical system and the random matrix theory is examined here by introducing a model of an autonomous system of two nonlinearly coupled spins. The proposed model can realize all three universality classes---the orthogonal, the unitary, and the symplectic---of Gaussian random matrices depending upon the nature of the nonlinearity. The proposed system evolves in a finite-dimensional Hilbert space which is in contrast with the existing models of autonomous systems requiring an infinite-dimensional Hilbert space for their description. The model is, therefore, not only free from the unpleasant problem of the truncation of the Hilbert space required for numerical work in the case of an infinite-dimensional Hilbert space but can also be used to examine for a dynamical system those aspects of the random matrix theory that are dependent on the dimension of the matrices. Those aspects are of particular interest in the Brownian motion theory of the transition from a Gaussian orthogonal ensemble to a Gaussian unitary ensemble.