Quantization of massive Dirac billiards and unification of nonrelativistic and relativistic chiral quantum scars

Quantum scars constitute one of the fundamental pillars in the traditional field of nonrelativistic quantum chaos. In relativistic quantum systems, chiral scars have been discovered recently whose wave functions concentrate on odd periodic orbits and break the time-reversal symmetry. We develop a theoretical framework to unify the scarring phenomena in nonrelativistic and relativistic quantum systems, which were previously thought to be distinct. In particular, we exploit massive Dirac billiard systems and derive semiclassical quantization rules to bridge the two opposite limits: the massless Dirac case and the large mass regime where the system effectively degenerates into one governed by the Schrödinger equation. A nontrivial phase is uncovered, which depends on the mass, the wave number, and the angle of reflection, and we demonstrate that this phase plays a key role in transforming the chiral scars and in bridging the relativistic and nonrelativistic quantum scars. In the large-mass limit, time-reversal symmetry is restored, as evidenced by a spectral analysis.