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

Abstract

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.

Perspectives on relativistic quantum chaos

“…Most of the understandings achieved so far for relativistic quantum chaos are for massless cases. Massive Dirac billiards have been considered only recently, where quantization formula for scarred states in confined 2D massive Dirac billiard has been proposed and validated numerically, and restoration of timereversal symmetry in the infinite mass limit has been unveiled [570]. However, there are many other issues to be understood in the massive Dirac billiards, e.g.…”

“…Recently we have developed quantization rule for scars in massive 2D Dirac billiards with infinite mass confinement [570]. Compare to the massless case, there is a new phase emerging during each reflection j.…”

“…), the mass m, and the wavenumber k (or energy E) [570]. Let g w = å   2 j j and in the absence of magnetic flux, i.e.…”

Electrical confinement in a spectrum of two-dimensional Dirac materials with classically integrable, mixed, and chaotic dynamics

“…Recurrence of quantum scars can also arise in graphene billiard systems [102]. In relativistic quantum billiard systems with T -symmetry breaking [103], chiral scars of massless spin-1/2 fermions for certain classes of periodic orbits can arise [104][105][106], which can recur with the energy or the wave vector. In open (scattering) systems with quasibound states, there are still relationships among the classical period orbits, the wave functions, and directional emission in nonrelativistic quantum systems, but the current understanding is that recurrence of the quasibound states is unlikely [37,107,108].…”

Scarring in classical chaotic dynamics with noise

“…Dismissed for a while as transients of no effect on the long-term properties of a closed chaotic system subject to thermalization [6], scars were brought back into the spotlight by recent numerical evidence of ergodicity breaking in manybody systems [7][8][9][10][11][12]. Further theoretical and experimental work has extended the notion of scarring to regular dynamics [13][14][15], to integrable systems with disorder [16][17][18], of interest in cold atoms and condensed matter, as well as to relativistic Dirac billiards [19,20].…”