具有经典可积台球形状的石墨烯台球中的GOE统计
GOE statistics in graphene billiards with the shape of classically integrable billiards
摘要：
量子混沌研究的一个重要结果是，经典可积系统的谱特性一般可用Poisson统计描述，而时间反演对称的经典混沌系统的谱特性与高斯正交系综（GOE）随机矩阵的谱特性一致。这个结果对二维狄拉克物质系统成立吗？为了解决这一基本问题，我们研究了一类具有代表性的石墨烯台球的光谱特性，其形状为经典可积的圆扇区台球。人们可能会天真地期望观察到泊松统计，这对于靠近能带边缘的能量确实是正确的，在那里准粒子服从薛定谔方程。然而，对于狄拉克点附近的能量，准粒子的行为就像无质量的狄拉克费米子，泊松统计是非常罕见的，因为它只出现在相当严格的对称性约束下的直边界部分的部门。任意少量的边界缺陷导致GOE统计。这意味着，对于具有任意角度的圆扇形约束，谱特性一般是GOE。这些结果得到了广泛的数值计算的证实。此外，我们还为我们的结果提供了物理理解。
英文摘要：
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 Schrödinger 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.
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