Non-Hermitian topological systems with eigenvalues that are always real

The effect of non-Hermiticity in band topology has sparked many discussions on non-Hermitian topological physics. It has long been known that non-Hermitian Hamiltonians can exhibit real energy spectra under the condition of parity-time (PT) symmetry-commonly implemented with balanced loss and gain-but only when non-Hermiticity is relatively weak. Sufficiently strong non-Hermiticity, on the other hand, will destroy the reality of energy spectra, a situation known as spontaneous PT-symmetry breaking. Here, based on nonreciprocal coupling, we show a systematic strategy to construct non-Hermitian topological systems exhibiting bulk and boundary energy spectra that are always real, regardless of weak or strong non-Hermiticity. Such nonreciprocal-coupling-based non-Hermiticity can directly drive a topological phase transition and determine the band topology, as demonstrated in a few non-Hermitian systems from one dimensional to two dimensional. Our work develops a theory that can guarantee the reality of energy spectra for non-Hermitian Hamiltonians, and offers an avenue to explore non-Hermitian topological physics.