Two-point correlations of the Gaussian symplectic ensemble from periodic orbits

Abstract

Abstract. We determine the asymptotics of the two-point correlation function for quantum systems with half-integer spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the leading terms of the two-point correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the rôle of Kramers' degeneracy.

Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

“…Here τ H stands for 1 for the orthogonal universality class and 2 for the symplectic class; such choice is equivalent to the replacement 2ε → ε in the symplectic case motivated by the Kramers degeneracy [11]. The semiclassical evaluation of the complex correlator is based on the generating function defined as the averaged ratio of four spectral determinants,…”

“…if Im ε > δ > 0. Early estimates of the oscillatory components are contained in [10,11]. Systematic approach is based on the formalism of generating functions, i.e., averaged ratios of the spectral determinants.…”

Level spacings and periodic orbits

“…. Although this assumption leads to the correct result in the case of two-point correlations [8,18], we remark that it corresponds to neglecting crosscorrelations between N fl T at different arguments, which, especially for small s, will become important. Thus, in the limit s → ∞, s/d → 0, the energy average in (6) can be performed approximately, yielding…”

“…This last step can be done by substituting the trace formula (1) for N fl T , employing an expansion in s/d (cf. [18]) and making use of the diagonal approximation [5,4],…”

The Gutzwiller Trace Formula for Quantum Systems with Spin

“…It is shown that, within the same range of validity as in [21], indeed the two-point correlations agree with the GSE or GUE, respectively. Moreover, in [24] the Bogomolny-Keating method is carried over to the case of spin 1/2, with the same findings. In both studies, apart from a chaotic translational motion, ergodic properties of the combined dynamics described below (19) are needed in order to obtain an agreement with the RMT predictions.…”