Periodic-orbit theory of universality in quantum chaos

Abstract

We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor K(τ ) as power series in the time τ . Each term τ n of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.

Semiclassical expansion of parametric correlation functions of the quantum time delay

“…in agreement with the first term in the expansion of the random matrix results, (12) and (18). Previous evaluations of the diagonal approximation of the form factor for the time delay, respectively its Fourier transform, were performed in [23,24].…”

“…The derivation of K v (τ ) for the spectral form factor can be found in [12,13]. We state here only those details that are relevant for the following calculations.…”

“…The correlations between different periodic orbits in open systems that are responsible for the off-diagonal terms in the expansion of the form factor have the same origin as in closed systems [8,11,12]. The general formalism for evaluating the correlations between orbits for arbitrary order has been developed in [11,12,13].…”

“…The general formalism for evaluating the correlations between orbits for arbitrary order has been developed in [11,12,13]. Long periodic orbits have many close self-encounters in which two or more stretches of an orbit are almost identical, possibly up to time reversal.…”

“…the second term in the Taylor expansion, was calculated for uniformly hyperbolic systems in [8,9]. The third order for general chaotic systems was obtained in [10], and all higher order terms in [11,12,13]. Similar methods have been applied to derive off-diagonal terms, for example, for the conductance [14,15], the GOE-GUE transition [16], the shot-noise [17] and parametric correlations [18,19].…”