石墨烯平均场模型的狄拉克锥
Dirac cones for a mean-field model of graphene
AI总结：
本文证明在解离区域和非简并性假设下，石墨烯约化 Hartree-Fock 理论在第一布里渊区顶点出现狄拉克点，并通过大尺度极限下的紧束缚模型分析表明费米能级恰好位于锥的汇合点。
摘要：
本文证明，在解离区域且满足非简并性假设的条件下，石墨烯的约化 Hartree-Fock 理论在第一布里渊区的顶点处呈现狄拉克点，且费米能级恰好位于锥的汇合点。为此，我们首先考虑作用于 $L^2(\mathbb{R}^2)$ 上的一般 Schrödinger 算子 $H=-Δ+V_L$，其中势 $V_L$ 关于某个具有长度尺度 $L$ 的晶格呈周期性。在涵盖周期性约化 Hartree-Fock 理论的若干假设下，我们证明在 $L\to\infty$ 极限下，$H_L$ 的低位谱带在主导阶上由紧束缚模型给出。对于石墨烯的六角晶格，后者在第一布里渊区的顶点处呈现奇异性。此外，Bloch 带的形状使得费米能级恰好位于锥上。
英文摘要：
In this article, we show that, in the dissociation regime and under a non-degeneracy assumption, the reduced Hartree-Fock theory of graphene presents Dirac points at the vertices of the first Brillouin zone and that the Fermi level is exactly at the coincidence point of the cones. For this purpose, we first consider a general Schrödinger operator $H=-Δ+V_L$ acting on $L^2(\mathbb{R}^2)$ with a potential $V_L$ which is assumed to be periodic with respect to some lattice with length scale $L$. Under some assumptions which covers periodic reduced Hartree-Fock theory, we show that, in the limit $L\to\infty$, the low-lying spectral bands of $H_L$ are given to leading order by the tight-binding model. For the hexagonal lattice of graphene, the latter presents singularities at the vertices of the Brillouin zone. In addition, the shape of the Bloch bands is so that the Fermi level is exactly on the cones.
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