Edge states and topological phases in non-Hermitian systems

Abstract

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.Comment: 29 pages, 19 figures, typos fixe

Exceptional points in a non-Hermitian topological pump

“…A number of recent works [13][14][15][16][17][18] have started to explore how band topological ideas might be extended to nonHermitian systems. Notably, certain special one-and two-dimensional non-Hermitian lattices have been found to exhibit topological invariants with meaningful bulkedge correspondences.…”

PT-breaking threshold in spatially asymmetric Aubry-André and Harper models: Hidden symmetry and topological states

“…Prima facie, these results predict the existence of topological insulators with a positive PT -breaking threshold [42,43] since our model makes no reference to the quantum statistics of the particle. In reality, however, amplification of a single degree of freedom is incompatible with the Pauli principle.…”

“…When β is irrational, the AAH model describes one-dimensional quasicrystals [37][38][39][40]. For an infinite lattice, when β is rational, the tunneling amplitude is periodic, and the corresponding AAH model has robust topological edge states [41] and is related to topological insulators [42][43][44][45]. We emphasize that a lattice with tunneling profile t k is not, in general, reflection symmetric [46].…”

Pseudo Hermitian Interactions in the Dirac Equation

“…In view of the fact that electrostatic fields alone cannot provide confinement of the electrons, there have been quite a number of works on exact solutions of the relevant Dirac equation with different magnetic field configurations, for example, square well magnetic barriers [3][4][5], non-zero magnetic fields in dots [6], decaying magnetic fields [7], solvable magnetic field configurations [8], etc. On the other hand, at the same time, there have been some investigations into the possible role of non-Hermiticity and PT symmetry [9] in graphene [10][11][12], optical analogues of relativistic quantum mechanics [13] and relativistic non-Hermitian quantum mechanics [14], photonic honeycomb lattice [15], etc. Furthermore, the (2 + 1)-dimensional Dirac equation with non-Hermitian Rashba and scalar interaction was studied [16].…”