$Z_2$ antiferromagnetic topological insulators with broken $C_4$ symmetry

TL;DR

This paper investigates $Z_2$ topological insulators in antiferromagnetic systems with broken $C_4$ symmetry, revealing limitations of existing invariants and proposing numerical methods to accurately identify topological phases.

Contribution

It demonstrates that previous $Z_2$ invariant expressions fail without certain lattice symmetries and introduces numerical techniques for diagnosing topological phases.

Findings

Existing $Z_2$ invariants do not detect all phases when $C_4$ symmetry is broken.

Numerical methods successfully identify topological phases in toy models.

Edge states and Chern numbers confirm the topological nature of phases.

Abstract

A two-dimensional topological insulator may arise in a centrosymmetric commensurate N\'{e}el antiferromagnet (AF), where staggered magnetization breaks both the elementary translation and time reversal, but retains their product as a symmetry. Fang et al.[Phys. Rev. B 88, 085406 (2013)] proposed an expression for a $Z_2$ topological invariant to characterize such systems. Here, we show that this expression does not allow to detect all the existing phases if a certain lattice symmetry is lacking. We implement numerical techniques to diagnose topological phases of a toy Hamiltonian, and verify our results by computing the Chern numbers of degenerate bands, and also by explicitly constructing the edge states, thus illustrating the efficiency of the method.