留数定理在拓扑相变中的应用

留数定理是高校物理专业必修课程数学物理方法中的一个重要定理.传统教学中关于该定理的讲授着重于数学公式的推导和数学思想的传达,而对于其在具体物理问题上的应用鲜有涉及.本文以一维Su-Schrieffer-Heeger模型的拓扑相变问题为例,阐明了如何利用留数定理解析得到二阶位移量的表达式并用该物理量表征拓扑相变.在讲授留数定理的教学过程中引入具体物理问题的分析实例,可以使学生更深刻地理解数学定理中的物理内涵.

Application of the residue theorem in topological phase transitions

The residue theorem is an important theorem in the mathematical physics methods, which is a compulsory course of physics major in college． The teaching of this theorem in traditional teaching focuses only on the derivation of mathematical formulas and the transmission of mathematics thoughts, but it is rarely involved in its application in specific physical problems． Taking the problem of the topological phase transition in one-dimensional Su-Schrieffer-Heeger model as an example, one explains how to use the residue theorem to analytically obtain the expression of the second-order displacement and use this physical quantity to characterize the topological phase transition． Introducing the analysis examples of specific physical problems in the teaching process of the residue theorem can enable students to understand the physical connotation of mathematical theorems more deeply．