Three-dimensional symmetry breaking topological matters

TL;DR

This paper explores three-dimensional topological states arising from symmetry-breaking, characterized by Chern numbers, including transitions from topological insulators to semimetals under Zeeman fields.

Contribution

It introduces a framework for understanding symmetry-breaking topological states in 3D systems using an extended Dirac Hamiltonian and Chern numbers.

Findings

Topological states can exist without fundamental symmetries.

Zeeman fields induce phase transitions from insulators to semimetals.

Chern number characterizes these symmetry-breaking topological phases.

Abstract

We discuss topological electronic states described by the Dirac Hamiltonian plus an additional one in three-dimension. When the additional Hamiltonian is an element of an Abelian group, electronic states become topologically nontrivial even in the absence of fundamental symmetries such as the time-reversal and the particle-hole symmety. The symmetry-breaking topological states are charercterized by the Chern number defined in the two-dimensional partial Brillouin zone. The topological insulators under Zeeman field are an example of the symmetry-breaking topological matters. We show the transision from the topological insulating phase to the topological semimetal one under the strong Zeeman field.