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

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.

[1]  G. Morandi,et al.  Path-integrals in multiply-connected spaces and the Aharonov-Bohm effect , 1984 .

[2]  D. Bohm,et al.  Significance of Electromagnetic Potentials in the Quantum Theory , 1959 .

[3]  R. Balian,et al.  Solution of the Schrodinger Equation in Terms of Classical Paths , 1974 .

[4]  Marko Robnik,et al.  Classical dynamics of a family of billiards with analytic boundaries , 1983 .

[5]  M. Berry,et al.  Semiclassical theory of spectral rigidity , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  O. Bohigas,et al.  Chaotic motion and random matrix theories , 1984 .

[7]  O. Bohigas,et al.  Characterization of chaotic quantum spectra and universality of level fluctuation laws , 1984 .

[8]  M. L. Mehta,et al.  On some Gaussian ensembles of Hermitian matrices , 1983 .

[9]  S. Olariu,et al.  The quantum effects of electromagnetic fluxes , 1985 .

[10]  Michael V Berry,et al.  Regular and irregular semiclassical wavefunctions , 1977 .

[11]  J. S. Dehesa,et al.  Mathematical and Computational Methods in Nuclear Physics , 1984 .

[12]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[13]  W. B. Joyce,et al.  Sabine’s reverberation time and ergodic auditoriums , 1975 .

[14]  A. Pandey.  Statistical properties of many-particle spectra. IV. New ensembles by Stieltjes transform methods☆ , 1981 .

[15]  M. Berry,et al.  Classical billiards in magnetic fields , 1985 .

[16]  Marko Robnik,et al.  Quantising a generic family of billiards with analytic boundaries , 1984 .

[17]  M. Berry,et al.  Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue , 1980 .

[18]  J. Verbaarschot,et al.  Quantum spectra of classically chaotic systems without time reversal invariance , 1985 .