Competing topological phases in a non-Hermitian time-reversal symmetry broken Bernevig-Hughes-Zhang model

The Bernevig-Hughes-Zhang (BHZ) model, a cornerstone in studying the quantum spin Hall insulators, showcases robust spin-filtered helical edge states in a nanoribbon geometry. In the presence of an in-plane magnetic field (IPMF), these (first-order) helical states gap out to be replaced by second-order corner states under suitable open boundary conditions. Here, we show that the inclusion of a spin-dependent non-Hermitian (NH) balanced gain/loss potential induces competition between these first- and second-order topological phases. Surprisingly, the previously dormant first-order helical edge states resurface as the NH effect intensifies, effectively neutralizing the role played by the magnetic field. By employing the projected spin spectra and the spin Chern number, we conclusively explain the resurgence of the first-order topological properties in the time-reversal symmetry-broken BHZ model in the presence of non-hermiticity. Notably, on entering the NH regime, the spectrum of the NH projected spin operator for the nanoribbon configuration further shows that the IPMF, in the second-order topological phase (γ≤ |B_y|), establishes channels that facilitate scattering between the up-spin and the down-spin states. This is the main reason for the appearance of a gap in the real energy spectrum of the nanoribbonlike configuration we discussed. Finally, the biorthogonal spin-resolved Berry phase, exhibiting a nontrivial winding, definitively establishes the topological nature of these revived edge states, emphasizing the dominance of non-hermiticity over the magnetic field. This nuanced balance underscores the potential of NH physics to manipulate and sustain topological phases of matter in environments where they would otherwise be destabilized by external perturbations.