Spectral form factor of hyperbolic systems: leading off-diagonal approximation

Abstract

The spectral fluctuations of a quantum Hamiltonian system with time-reversal symmetry are studied in the semiclassical limit by using periodic-orbit theory. It is found that, if long periodic orbits are hyperbolic and uniformly distributed in phase space, the spectral form factor K(τ ) agrees with the GOE prediction of random-matrix theory up to second order included in the time τ measured in units of the Heisenberg time (leading off-diagonal approximation). Our approach is based on the mechanism of periodic-orbit correlations discovered recently by Sieber and Richter [1]. By reformulating the theory of these authors in phase space, their result on the free motion on a Riemann surface with constant negative curvature is extended to general Hamiltonian hyperbolic systems with two degrees of freedom.PACS numbers: 05.45.Mt, 03.65.Sq Hyperbolic Hamiltonian systemsWe consider a particle moving in a Euclidean plane (f = two degrees of freedom), with Hamiltonian H(q, p) = H(q, −p) invariant under time-reversal symmetry. We assume the existence of a compact two-dimensional Poincaré surface of section Σ in the (four-dimensional) phase space Γ, contained in an energy shell H(q, p) = E and invariant under time reversal (TR) [4,12]. Every classical orbit of energy E intersects Σ transversally. The classical dynamics can then be described by an area-preserving map φ on Σ, together with a first-return time map x ∈ Σ → t x ∈ [0, ∞] (see [12]). In what follows, letters in normal and bold fonts are assigned

Field theory approach to quantum interference in chaotic systems

“…Yet in spite of its ubiquity, and notwithstanding a number of significant recent advances [5,6,7,8,9,10,11], the correspondence above is not yet fully understood theoretically. Specifically, the 'non-perturbative' aspects of the problem -which manifest themselves, e.g., in the low energy profile of spectral correlations -are not under quantitative control.…”

Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. II. Semiclassical approach

“…In the form factor these Sieber-Richter pairs give the quadratic order in time as predicted by the GOE. Their approach has been generalized to general hyperbolic billiards [7] and later in a phase space approach * Electronic address: sven@gnutzmann.de † Electronic address: bseif@thp.uni-koeln.de to hyperbolically chaotic Hamiltonian systems with two [8,9] and finally any number of degrees of freedom [10]. Finally, the cubic order in the short-time expansion has been calculated [11].…”

Universal spectral form factor for chaotic dynamics

“…Each Sieber-Richter (SR) pair has a close self-encounter which in configuration space looks like a small-angle crossing for one orbit and like a narrowly avoided crossing for the partner orbit. Generalizations to arbitrary hyperbolic systems with two freedoms were given in [8][9][10] and for more freedoms in [11].…”