GOE statistics in graphene billiards with the shape of classically integrable billiards

TL;DR

This paper investigates the spectral statistics of graphene billiards with integrable shapes, revealing that near the Dirac point, the statistics are predominantly GOE due to boundary imperfections, contrasting with classical expectations.

Contribution

It demonstrates that in graphene billiards of integrable shape, spectral statistics near the Dirac point are typically GOE, challenging classical quantum chaos predictions for such systems.

Findings

Spectral statistics near the band edges follow Poisson distribution.

Near the Dirac point, spectral statistics are mostly GOE due to boundary imperfections.

Boundary imperfections strongly influence the spectral behavior of graphene billiards.

Abstract

A crucial result in quantum chaos, which has been established for a long time, is that the spectral properties of classically integrable systems generically are described by Poisson statistics whereas those of time-reversal symmetric, classically chaotic systems coincide with those of random matrices from the Gaussian orthogonal ensemble (GOE). Does this result hold for two-dimensional Dirac material systems? To address this fundamen- tal question, we investigate the spectral properties in a representative class of graphene billiards with shapes of classically integrable circular-sector billiards. Naively one may expect to observe Poisson statistics, which is indeed true for energies close to the band edges where the quasiparticle obeys the Schr\"odinger equation. However, for energies near the Dirac point, where the quasiparticles behave like massless Dirac fermions, Pois- son statistics is extremely rare in the sense that it emerges only under quite strict symmetry constraints on the straight boundary parts of the sector. An arbitrarily small amount of imperfection of the boundary results in GOE statistics. This implies that, for circular sector confinements with arbitrary angle, the spectral properties will generically be GOE. These results are corroborated by extensive numerical computation. Furthermore, we provide a physical understanding for our results.