  In general, the energy spectrum of a non-Hermitian system turns out to be
complex, which is not so satisfactory since the time evolution of eigenstates
with complex eigenvalues is either exponentially growing or decaying. Here we
provide a sufficient and necessary condition of the real spectrum under open
boundary conditions for one-dimensional non-Hermitian tight-binding
Hamiltonians. The necessity is directly related to the fact that the
generalized Brillouin zone in one dimension is a closed loop with the origin in
its interior. For some simple models, we demonstrate the sufficiency by
analytically determining when the preimage of the characteristic polynomial
contains a loop and showing that this loop is just the generalized Brillouin
zone itself. For more complex models, the numerical results are shown. Our
results indicate that real spectra are more common than one may have expected
in non-Hermitian systems and are helpful for designing non-Hermitian models
with real spectra.
