School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA.
Chaos. 2011 Mar;21(1):013102. doi: 10.1063/1.3537814.
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.
在非相对论量子非线性动力学中，一个基本的结果是，对于没有几何对称性的量子系统，但其经典动力学是混沌的，在存在或不存在时间反演对称性的情况下，其谱统计分别由高斯正交系综（GOE）或高斯幺正系综（GUE）描述。对于无质量的自旋-半粒子，如相对论量子力学中的中微子在混沌 billiard 中，Berry 和 Mondragon 的开创性工作确立了能级间距统计的 GUE 性质，这是由于狄拉克粒子的手性和限制的结合，破坏了时间反演对称性。一个问题是，在实验可及的相对论量子系统中是否可以观察到 GOE 或 GUE 统计。我们使用石墨烯限制来证明，在低能区，准粒子运动由狄拉克方程控制，能级间距统计始终是 GOE 随机矩阵的统计。我们从紧束缚方法中获得了广泛的数值证据，并对 GOE 统计进行了物理解释。我们还发现，弱磁场的存在会将统计切换到 GUE。对于强磁场，朗道能级变得有影响，导致能级间距分布明显偏离随机矩阵预测。还讨论了一些实际因素对能级间距统计的影响，如最近邻相互作用、不同的晶格取向、边界原子的跳跃能量增强以及由于石墨烯-衬底相互作用引起的交错势。
