### Abstract

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3537814]

Original language | English |
---|---|

Article number | 013102 |

Number of pages | 11 |

Journal | Chaos |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2011 |

### Keywords

- time-reversal symmetry
- carbon nanotubes
- magnetic-fields
- energy-levels
- quantum dots
- phase

### Cite this

*Chaos*,

*21*(1), [013102]. https://doi.org/10.1063/1.3537814

**Characteristics of level-spacing statistics in chaotic graphene billiards.** / Huang, Liang; Lai, Ying-Cheng; Grebogi, Celso.

Research output: Contribution to journal › Article

*Chaos*, vol. 21, no. 1, 013102. https://doi.org/10.1063/1.3537814

}

TY - JOUR

T1 - Characteristics of level-spacing statistics in chaotic graphene billiards

AU - Huang, Liang

AU - Lai, Ying-Cheng

AU - Grebogi, Celso

PY - 2011/3

Y1 - 2011/3

N2 - A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3537814]

AB - A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3537814]

KW - time-reversal symmetry

KW - carbon nanotubes

KW - magnetic-fields

KW - energy-levels

KW - quantum dots

KW - phase

UR - http://www.scopus.com/inward/record.url?scp=79957546606&partnerID=8YFLogxK

U2 - 10.1063/1.3537814

DO - 10.1063/1.3537814

M3 - Article

VL - 21

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 1

M1 - 013102

ER -