Gaussian orthogonal ensemble statistics in graphene billiards with the shape of classically integrable billiards

Pei Yu, Zi-Yuan Li, Hong-Ya Xu , Liang Huang, Barbara Dietz, Celso Grebogi, Ying-Cheng Lai

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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 fundamental 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 Schrodinger equation. How- ¨
ever, for energies near the Dirac point, where the quasiparticles behave like massless Dirac fermions, Poisson 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.
Original languageEnglish
Article number062214
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Issue number6
Publication statusPublished - 20 Dec 2016

Bibliographical note

L.H. thanks Professor A. Rycerz for helpful discussions. This work was supported by NSF of China under Grants No. 11135001, No. 11375074, and No. 11422541, as well as by the Doctoral Fund of Ministry of Education of China under Grant No. 20130211110008. Y.C.L. and H.Y.X. were supported by AFOSR under Grant No. FA9550-15-1-0151 and by ONR under Grant No. N00014-15-1-2405.


  • quant-ph
  • nlin.CD


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