Origin of Robust ${\mathbb{Z}}_2$ Topological Phases in Stacked Hermitian Systems: Non-Hermitian Level Repulsion

Abstract

Quantum spin Hall insulators, which possess a non-trivial ${\mathbb{Z}}_2$ topological phase, have attracted great attention for two decades. It is generally believed that when an even number of layers of the quantum spin Hall insulators are stacked, the ${\mathbb{Z}}_2$ topological phase becomes unstable due to ${\mathbb{Z}}_2$ nature. While the counterexamples of the instability were observed in several literates, there is no systematic understanding. In this work, we provide a systematic understanding that the robust ${\mathbb{Z}}_2$ topological phase in a Hermitian system with chiral symmetry against stacking. We clarify that the robustness generally originates from level repulsion in the corresponding non-Hermitian system derived from Hermitization. We demonstrate this by treating a class DIII superconductor in 1D with ${\mathbb{Z}}_2$ topology and the corresponding non-Hermitian 1D system in class AII${}^{†}$ with ${\mathbb{Z}}_2$ point-gap topology.

13 pages, 11 figures

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