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

TL;DR

This paper demonstrates that non-Hermitian PT-symmetric Hamiltonians do not support topological insulator phases, as such states require PT-breaking and have complex spectra, challenging their physical viability.

Contribution

It shows that topological insulator states cannot be realized in non-Hermitian PT-symmetric Hamiltonians, contrasting with Hermitian cases.

Findings

Topological insulator states are PT-breaking phases.

Non-Hermitian PT-symmetric Hamiltonians have complex energy spectra.

Such Hamiltonians are not consistent quantum theories.

Abstract

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 bulk and boundary state proofs 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 which are complex (not real) and thus such non-Hermitian Hamiltonians are not consistent quantum theories.