  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 non-reciprocal
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 1D to
2D. Our work develops so far the only theory that can guarantee the reality of
energy spectra for non-Hermitian Hamiltonians, and offers a new avenue to
explore non-Hermitian topological physics.
