Absence of topological insulator phases in non-Hermitian $PT$-symmetric Hamiltonians

In this work, we consider a generalization of the symmetry classification of topological insulators to non-Hermitian Hamiltonians, which satisfy a combined $PT$ symmetry (parity and time reversal). We show via examples and explicit proofs from separate bulk and gapless boundary-state perspectives that the typical paradigm of forming topological insulator states from Dirac Hamiltonians is not compatible with the construction of non-Hermitian $PT$-symmetric Hamiltonians. The topological insulator states are $PT$-breaking phases and have energy spectra that are complex (not real) and, thus, are not consistent quantum theories.

Figure 1

Energy spectrum of $H_{1\text{D},\text{lattice}}^{\prime }$ [Eq. (4)] with open boundaries vs $m$. (a) $\Delta =0$ gives a real spectrum with zero modes for $-2<m<0.$ (b) Real part of energy spectrum and (c) imaginary part of energy spectrum for $\Delta =0.1.$ Complex eigenvalues exist in the former topological phase.