Relativistic quantum scarring, spin-induced phase, and quantization in a symmetric Dirac billiard system

Thirty-five years ago, Sir Michael Berry and his collaborator Mondragon studied the behaviors of neutrino, a massless relativistic quantum particle, in a classically chaotic billiard-the neutrino billiard problem. To celebrate Sir Michael Berry's eightieth birthday, here we report results on the role of geometric symmetries of the billiard system in relativistic quantum scarring. In particular, we investigate a Dirac billiard system with a four-fold rotational symmetry whose classical dynamics are fully chaotic. The system is described by the massless Dirac equation in the fundamental domain that consists of one fourth of the full billiard, with proper boundary conditions on the symmetry lines to preserve the physical properties under the symmetry operations. We show that the relativistic quantum characteristics of spin induced phase play a fundamental role in the quantum behaviors of the Dirac particle in the billiard. We find that the peaks in the length spectra are due to the interference of states circling the fundamental domain orbits (FDOs) in opposite propagating directions, which can be constructive or destructive depending on the accumulated phases. In addition, we derive the quantization conditions of the scarring states about the unstable periodic orbits within the fundamental domain from the phase along the FDOs. Our work is a vivid demonstration that relativistic quantum scarring, or more generally quantum manifestations of classical chaos, can be fully understood by analyzing the behaviors of the geometric phase-a powerful approach in modern physics pioneered by Sir Michael Berry.