Origin of Robust mathbb{Z}₂ Topological Phases in Stacked Hermitian Systems: Non-Hermitian Level Repulsion

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^dagger with mathbb{Z}₂ point-gap topology.