Absence of topological insulator phases in non-HermitianPT-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.