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