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Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians. As an SU(1,1) Hamiltonian, the tight-binding model on the honeycomb lattice with imaginary on-site potentials is examined. Edge states with ReE=0 and their topological stability are discussed by the winding number and the index theorem, based on the pseudo-anti-Hermiticity of the system. As a higher symmetric generalization of SU(1,1) Hamiltonians, we also consider SO(3,2) models. We investigate non-Hermitian generalization of the Luttinger Hamiltonian on the square lattice, and that of the Kane-Mele model on the honeycomb lattice, respectively. Using the generalized Kramers theorem for the time-reversal operator Theta with Theta^2=+1 [M. Sato et al., arXiv:1106.1806], we introduce a time-reversal invariant Chern number from which topological stability of gapless edge modes is argued.
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Far from being limited to a trivial generalization of their Hermitian counterparts, non-Hermitian topological phases have gained widespread interest due to their unique properties. One of the most striking non-Hermitian phenomena is the skin effect, i.e., the localization of a macroscopic fraction of bulk eigenstates at a boundary, which underlies the breakdown of the bulk-edge correspondence. Here we investigate the emergence of the skin effect in magnetic insulating systems by developing a phenomenological approach to describing magnetic dissipation within a lattice model. Focusing on a spin-orbit-coupled van der Waals (vdW) ferromagnet with spin-nonconserving magnon-phonon interactions, we find that the magnetic skin effect emerges in an appropriate temperature regime. Our results suggest that the interference between Dzyaloshinskii-Moriya interaction (DMI) and nonlocal magnetic dissipation plays a key role in the accumulation of bulk states at the boundaries.
The hallmark of symmetry-protected topological (SPT) phases is the existence of anomalous boundary states, which can only be realized with the corresponding bulk system. In this work, we show that for every Hermitian anomalous boundary mode of the ten Altland-Zirnbauer classes, a non-Hermitian counterpart can be constructed, whose long time dynamics provides a realization of the anomalous boundary state. We prove that the non-Hermitian counterpart is characterized by a point-gap topological invariant, and furthermore, that the invariant exactly matches that of the corresponding Hermitian anomalous boundary mode. We thus establish a correspondence between the topological classifications of $(d+1)$-dimensional gapped Hermitian systems and $d$-dimensional point-gapped non-Hermitian systems. We illustrate this general result with a number of examples in different dimensions. This work provides a new perspective on point-gap topological invariants in non-Hermitian systems.
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