A note on the topological insulator phase in non-Hermitian quantum systems.

Examples of non-Hermitian quantum systems admitting a topological insulator phase are presented in one, two and three space dimensions. All of these non-Hermitian Hamiltonians have entirely real bulk eigenvalues and unitarity is maintained with the introduction of appropriate inner products in the corresponding Hilbert spaces. The topological invariant characterizing a particular phase is shown to be identical for a non-Hermitian Hamiltonian and its Hermitian counterpart, to which it is related through a non-unitary similarity transformation. A classification scheme for topological insulator phases in pseudo-Hermitian quantum systems is suggested.