Constructing Hopf insulator from geometric perspective of Hopf invariant

Abstract

Abstract We propose a method to construct Hopf insulators based on the study of topological defects from the geometric perspective of Hopf invariant $I$. Firstly, we prove two types of topological defects naturally inhering in the inner differential structure of the Hopf mapping. One type is the four-dimensional (4D) point defects, which lead to a topological phase transition occurs at the Dirac points. The other type is the three-dimensional (3D) merons, whose topological charges give the evaluations of $I$. Then, we show two ways to establish the Hopf insulator models. One approach is to modify the locations of merons, thereby the contributions of charges to $I$ will change. The other approach is related to the number of defects. It is discovered that $I$ will decrease if the number reduces, while increase if additional defects are added. The method developed in this paper is expected to provides a new perspective for understanding the topological invariants, which opens a new door in exploring and designing novel topological materials in three dimensions.