Statistics of energy levels without time-reversal symmetry: Aharonov-Bohm chaotic billiards

Abstract

A planar domain D contains a single line of magnetic flux Phi . Switching on Phi breaks time-reversal symmetry (T) for quantal particles with charge q moving in D, whilst preserving the geometry of classical (billiard) trajectories bouncing off the boundary delta D. If delta D is such that these classical trajectories are chaotic, the authors predict that T breaking will cause the local statistics of quantal energy levels to change their universality class, from that of the Gaussian orthogonal ensemble (GOE) of random-matrix theory to that of the Gaussian unitary ensemble (GUE). In the semiclassical limit this transition is abrupt; for statistics involving the first N levels, GUE behaviour requires that the quantum flux alpha identical to q Phi /h>>0.13N-14/. The special flux alpha =1/2 corresponds to 'false T breaking' and for this case GOE statistics are predicted. These predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry, for which delta D is a cubic conformal image of the unit disc.