Chaotic dynamics and the GOE-GUE transition

The statistical fluctuations of the energy levels of classically chaotic systems are believed to coincide with those of Gaussian matrix ensembles. The transition from the Gaussian orthogonal ensemble (GOE) to the Gaussian unitary ensemble (GUE), as the time-reversal symmetry is broken, is known exactly. The asymptotic form of the two-point correlation function is derived from semiclassical periodic orbit theory, leading to a dynamical evaluation of the transition parameter. Numerical calculations of the correlation function for a chaotic billiard in a constant magnetic field reveal a clear crossover from GUE behaviour to GOE as the level separation is increased, in agreement with the theoretical prediction.