False time-reversal violation and energy level statistics: the role of anti-unitary symmetry

Abstract

The authors extend the classification of symmetries necessary to predict the universality class of spectral fluctuations of quantal systems whose classical motion is chaotic, by explaining that a system with neither time-reversal symmetry (T) nor geometric symmetry may display the spectral statistics of the Gaussian orthogonal ensemble (GOE), rather than those of the Gaussian unitary ensemble (GUE), provided it possesses instead some combination of symmetries which includes T. Such combinations constitute invariance under anti-unitary transformations (whose classical analogue are called anticanonical). For a particle in a magnetic field B plus scalar potential V, an example is TSx where Sx is a mirror reflection under which B and V are invariant. The authors illustrate this numerically for a single flux line in a hard-walled enclosure (Aharonov-Bohm quantum billiards), which also provides an example of an anti-unitary symmetry of non-geometrical origin; the spectral fluctuations are, as predicted, GOE rather than GUE.