Tight-binding reduction and topological equivalence in strong magnetic fields
Tight-binding reduction and topological equivalence in strong magnetic fields
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强磁场中的紧束缚约简和拓扑等价
DOI：
10.1016/j.aim.2022.108343
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发表时间：
2020
影响因子：
1.7
通讯作者：
M. Weinstein
中科院分区：
文献类型：
--
作者：
Jacob Shapiro;M. Weinstein
拓扑绝缘体（TI）是一类材料，其在其本体形式下是绝缘的，但是在引入边界或边缘时，例如通过突然终止材料，可以沿着其边界沿着表现出自发电流。该属性通过与主体或边缘系统相关联的拓扑指数来量化。在凝聚态物理学领域，紧束缚（离散）近似模型，参数化跳跃系数，已成功地用于捕获的拓扑行为的TI在许多设置。然而，这样的紧束缚模型是否捕捉到了与量子物理学的基础连续模型相同的拓扑特征一直是一个悬而未决的问题。我们解决这个问题的背景下，拓扑行为的材料，整数量子霍尔效应的原型例子。本文研究了在垂直磁场作用下二维晶体中电子运动的一类连续哈密顿量H λ。对晶体的平移不变性没有任何假设。我们证明了，在磁场强度和晶体势的深度都足够大的情况下，λ <$1，H λ的低能谱和本征态（以及相应的大时间动力学）可以用无标度离散哈密顿量H TB很好地描述;我们证明了范数预解式收敛。相关的拓扑指数是霍尔电导率，其可表示为Fredholm指数。我们证明了当λ较大时，H λ和HTB的拓扑指数一致.这是证明散装和边缘的几何形状分别。我们的结果证明了在拓扑物质的研究中使用离散模型的原则。
Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, eg 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≫ 1, 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 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 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.
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影响因子：
2.4
作者：
A. Drouot
通讯作者：
A. Drouot
影响因子：
1.7
作者：
A. Drouot;M. Weinstein
通讯作者：
A. Drouot;M. Weinstein
影响因子：
2.4
作者：
C. Fefferman;M. Weinstein
通讯作者：
C. Fefferman;M. Weinstein
DOI：
10.1088/1751-8121/ac16c4
发表时间：
2021
期刊：
Journal of Physics A: Mathematical and Theoretical
影响因子：
--
作者：
Simon, Becker;Han, Rui;Jitomirskaya, Svetlana;Zworski, Maciej
通讯作者：
Zworski, Maciej