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Evidence for the suppression of the hybrid skin-topological effect by fragile topology

Topological insulators are well-known for their topological edge states, which are protected by the non-trivial bulk topology and characterized by gapless Wannier bands, a phenomenon known as the bulk-boundary correspondence. However, fragile topology challenged this concept, the Wannier bands are gapped, but the edge states still exist with similar protection. Previous studies on fragile topology have primarily focused on the spectral flow under twisted boundary conditions, but the discussion on the physical interpretation of the Wannier gap is limited. In this study, we introduce a bilayer breathing honeycomb lattice with spiral interlayer couplings inside the unit cell. As we increase the interlayer coupling strength, the Wannier gap increases monotonically and the bandgap first increases then decreases. After introducing a gain-loss domain wall, the hybrid skin-topological effect (HSTE) emerges, and the topological edge states under the periodic boundary condition (PBC) change into corner states under the open boundary condition (OBC) associated with the significant spectral difference. HSTE is suppressed as the interlayer coupling strength increases, the spectral difference between the two boundary conditions has an overall decreasing trend, which more closely mirrors the evolution of the inverse of the Wannier gap. Moreover, some of the corner states transform into edge states. Our work first provides evidence for the relation between fragile topology and HSTE, shedding new insights into the underlying mechanism of Non-Hermitian skin effect (NHSE).