Beyond symmetry protection: Robust feedback-enforced edge states in non-Hermitian stacked quantum spin Hall systems

Abstract: Conventional wisdom holds that strongly coupling two QSH layers yields a trivial $\mathbb{Z}_2$ phase and no protected topological edge states. We demonstrate that, in a regime with intermediate inter-layer coupling (neither in the strong or weak coupling regimes) and competitive non-Hermitian directed amplification, bulk modes are suppressed while arbitrary bulk excitations inevitably accumulate into robust helical edge transport modes - without relying on any symmetry protection. Our feedback-enforced mechanism persists over broad parameter ranges and remains robust even on fractal or irregular boundaries. These findings challenge the traditional view of stacked QSH insulators as inevitably trivial, and open up new avenues for designing helical topological devices that exploit feedback-enforced non-Hermitian engineering, instead of symmetry-enforced robustness.