On the ℤ2 topological invariant

Abstract

We develop a complex fermionic field-based method to model the properties of the filled bands of topological two-dimensional (2D) matter with time reversal (TR)-symmetry. Using this fermionic representation, we give an explicit calculation of the [Formula: see text] index for 2D topological matter invariant under TR and comment on the emergence of Majorana states at the TR-fix points. Moreover, motivated by recent theoretical results on possible signatures of topological supersymmetric matter, we also give the supersymmetric generalization of our TR-invariant construction and calculate the underlying topological [Formula: see text] index. Other features such as the topological obstruction of basis sections in the fermionic determinant bundle are also investigated. Applications for the calculations of the supersymmetric charge [Formula: see text] operator and the super-Hamiltonian [Formula: see text] for the three-dimensional topological class AII are undertaken; these operators are given by Eqs. (5.48)–(5.51).