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We explore band structures of one-dimensional open systems described byperiodic non-Hermitian operators, based on continuum models and tight-bindingmodels. We show that imaginary scalar potentials do not open band gaps butinstead lead to the formation of exceptional points as long as the strength ofthe potential exceeds a threshold value, which is contrast to closed systemswhere real potentials open a gap with infinitesimally small strength. Theimaginary vector potentials hinder the separation of low energy bands becauseof the lifting of degeneracy in the free system. In addition, we constructtight-binding models through bi-orthogonal Wannier functions based on Blochwavefunctions of the non-Hermitian operator and its Hermitian conjugate. Weshow that the bi-orthogonal tight-binding model well reproduces the dispersionrelations of the continuum model when the complex scalar potential issufficiently large.

Band structures under non-Hermitian periodic potentials: Connecting nearly-free and bi-orthogonal tight-binding models