On Breaking Time Reversal in a Simple, Smooth, Chaotic System
Within random matrix theory, the statistics of the eigensolutions depend fundamentally on the presence (or absence) of time reversal symmetry. Accepting the Bohigas-Giannoni-Schmit conjecture, this statement extends to quantum systems with chaotic classical analogs. For practical reasons, much of the supporting numerical studies of symmetry breaking have been done with billiards or maps, and little with simple, smooth systems. There are two main difficulties in attempting to break time reversal invariance in a continuous time system with a smooth potential. The first is avoiding false symmetry breaking. The second is locating a parameter regime in which the symmetry breaking is strong enough to transition the fluctuation properties fully toward the broken symmetry case, and yet remain weak enough so as not to regularize the dynamics sufficiently that the system is no longer chaotic. We give an example of a system of two-coupled quartic oscillators whose energy level statistics closely match those of the Gaussian unitary ensemble, and which possesses only a minor proportion of regular motion in its phase space.