Origin of Robust Z₂ Topological Phases in Stacked Hermitian Systems: Non-Hermitian Level Repulsion

Quantum spin Hall insulators, which possess a non-trivial Z₂ 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 Z₂ topological phase becomes unstable due to Z₂ nature. While several researchers report counterexamples of the instability, there is no systematic understanding. In this work, we show that the Z₂ topological phase is robust against the stacking if it has an additional chiral symmetry, in terms of the level repulsion in the corresponding non-Hermitian system. We demonstrate this by mapping a one-dimensional class DIII superconductor with Z₂ topology to the corresponding non-Hermitian system in AII^ with Z₂ point-gap topology.