论文标题
强磁场的紧密结合减少和拓扑等效性
Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields
论文作者
论文摘要
拓扑绝缘子（TIS）是一类材料，它们以其大体形式进行绝缘，但在引入A边界或边缘时，例如通过突然终止材料，可以沿其边界表现出自发的电流。该特性由与散装系统或边缘系统相关的拓扑指数量化。在冷凝物理物理学领域中，通过跳跃系数进行了参数的紧密结合（离散）近似模型已成功地用于捕获许多情况下TI的拓扑行为。但是，这种紧密的结合模型是否捕获与量子物理的基础连续体模型相同的拓扑特征，这是一个悬而未决的问题。 我们在材料中拓扑行为的原型示例（整数量子厅效应）的原型示例中解决了这个问题。我们研究了一类连续汉密尔顿，$ h^λ$，该$ h^λ$在垂直磁场的影响下控制二维晶体中的电子运动。没有对晶体的翻译不变性做出的假设。我们证明，在磁场强度和晶体电位的深度都足够大的情况下，$λ\ gg1 $，低洼的能源谱和特征（以及相应的大型动力学）的$ h^λ$被很好地描述了一个无尺度的汉密尔顿，$ h^^$ h^$ h^$ h^$ rm tb};我们显示标准分解收敛。相关的拓扑指数是霍尔电导率，该指数可作为弗雷德尔姆指数表达。我们证明，对于大$λ$，$ h^λ$和$ h^{\ rm tb} $的拓扑索引对于散装和边缘几何形状，这是分别证明的。我们的结果证明了在拓扑问题研究中使用离散模型的原则是合理的。
Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, e.g. by abruptly terminating the material, may exhibit spontaneous current along their boundary. This property is quantified by topological indices associated with either the bulk or the edge system. In the field of condensed matter physics, tight binding (discrete) approximate models, parametrized by hopping coefficients, have been used successfully to capture the topological behavior of TIs in many settings. However, whether such tight binding models capture the same topological features as the underlying continuum models of quantum physics has been an open question. We resolve this question in the context of the archetypal example of topological behavior in materials, the integer quantum Hall effect. We study a class of continuum Hamiltonians, $H^λ$, which govern electron motion in a two-dimensional crystal under the influence of a perpendicular magnetic field. No assumption is made on translation invariance of the crystal. We prove, in the regime where both the magnetic field strength and depth of the crystal potential are sufficiently large, $λ\gg1$, that the low-lying energy spectrum and eigenstates (and corresponding large time dynamics) of $H^λ$ are well-described by a scale-free discrete Hamiltonian, $H^{\rm TB}$; we show norm resolvent convergence. The relevant topological index is the Hall conductivity, which is expressible as a Fredholm index. We prove that for large $λ$ the topological indices of $H^λ$ and $H^{\rm TB}$ agree. This is proved separately for bulk and edge geometries. Our results justify the principle of using discrete models in the study of topological matter.
