标题： 连续拓扑绝缘体分类与体边对应关系 显示英文标题

标题： Continuous Topological Insulators Classification and Bulk Edge Correspondence

摘要： 本文综述了在欧几里得空间中对描述体和界面拓扑绝缘体的偏微分算子分类的最新结果。 我们的主要目标是对分离不同相位的拓扑绝缘体的界面处必然出现的、对扰动具有鲁棒性的非对称输运进行数学分析。 分析的核心元素是一个描述这种不对称性的界面电流可观测量。 我们表明，当界面哈密顿量可以显式对角化时，该可观测量可以通过谱流来显式计算。 我们回顾了朗道算子和狄拉克算子的体相分类，并通过域墙和相应的体差不变量（BDI）对椭圆界面伪微分算子进行了普遍分类。 BDI可通过实现欧几里得设置中的阿蒂亚-辛格指标理论的费多索夫-霍尔曼公式进行简单计算。 然后，广义的体边对应关系指出，在椭圆算子上，界面电流可观测量和BDI是一致的，而对于非椭圆算子则不一定如此。 显示英文摘要

摘要： This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface current observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-H\"ormander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operator, whereas this is not necessarily the case for non-elliptic operators.