Corner modes in non-Hermitian next-nearest-neighbor hopping model

We consider a non-Hermitian (NH) analog of a second-order topological insulator, protected by chiral symmetry, in the presence of next-nearest-neighbor hopping elements to theoretically investigate the interplay beyond the first-nearest-neighbor hopping amplitudes and topological order away from Hermiticity. In addition to the four zero-energy corner modes present in the first-nearest-neighbor hopping model, we uncover that the second-nearest-neighbor hopping introduces another topological phase with 16 zero-energy corner modes. Importantly, the NH effects are manifested in altering the Hermitian phase boundaries for both of the models. While comparing the complex energy spectrum under open boundary conditions, and bi-orthogonalized quadrupolar winding number in real space, we resolve the apparent anomaly in the bulk boundary correspondence of the NH system as compared to the Hermitian counterpart by incorporating the effect of the non-Bloch form of momentum into the mass term. The above invariant is also capable of capturing the phase boundaries between the two different topological phases where the degeneracy of the corner modes is evident, as exclusively observed for the second-nearest-neighbor model.