Edge states and topological phases in non-Hermitian systems

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 onsite potentials is examined. Edge states with Re$E=0$ 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 $\ensuremath{\Theta}$ with ${\ensuremath{\Theta}}^{2}=+1$ [M. Sato et al., e-print arXiv:1106.1806], we introduce a time-reversal-invariant Chern number from which topological stability of gapless edge modes is argued.