Stadium in a Magnetic Field: Time-Reversal Invariance Symmetry Breaking and Energy Level Statistics

We investigate the breaking of time-reversal invariance for the energy spectra of nanostructure devices in the ballistic regime, by conducting numerical studies on a stadium billiard model described by a tight-binding Hamiltonian. The symmetry breaking is controlled by a uniform magnetic field, and for different sizes of systems a gradual transition from GOE behaviour to GUE behaviour is observed. The results confirm the qualitative criterion for the transition, namely that the transition field Bc corresponds to one flux quantum passing through the system. Furthermore, all the transition curves show universal behaviour when plotted against the flux through the system. Our results are in good agreement with the predictions of Random Matrix Theory (RMT), even though our Hamiltonian leads to matrices with a very different structure.