Topological Phase Transition without Gap Closing

TL;DR

This paper demonstrates that topological phase transitions can occur without gap closing by changing system symmetries, challenging the conventional belief that gap closing is necessary for topological changes.

Contribution

It introduces generic principles and examples showing topological transitions without gap closing, emphasizing symmetry changes and detours in parameter space.

Findings

Topological states can be connected without gap closing when symmetry changes.

Examples include 1D polyacetylene, 2D silicene, and 3D superconductors.

Topological indices can be ill-defined during the transition.

Abstract

Topological phase transition is accompanied with a change of topological numbers. It has been believed that the gap closing and the breakdown of the adiabaticity at the transition point is necessary in general. However, the gap closing is not always needed to make the topological index ill-defined. In this paper, we show that the states with different topological numbers can be continuously connected \textit{without} gap closing in some cases, where the symmetry of the system changes during the process. Here we propose the generic principles how this is possible (impossible) by demonstrating various examples such as 1D polyacetylene with the charge-density-wave order, 2D silicene with the antiferromagnetic order, 2D silicene or quantum well made of HgTe with superconducting proximity effects and 3D superconductor Cu doped Bi$_{2}$Se$_{3}$. It is argued that such an unusual phenomenon can occur when we detour around the gap closing point provided the connection of the topological indices is lost along the detour path.