Topology of the Generalized Brillouin Zone of One-Dimensional Models

Generalized Brillouin zones (GBZs) are integral in the analysis of non-Hermitian band structures. Conventional wisdom suggests that the GBZ should be connected, where each point can be indexed by the real part of the wave vector, similar to the Brillouin zone. Here we demonstrate rich topological features of the GBZs in generic non-Hermitian one-dimensional models. We prove and discuss a set of sufficient conditions for a single-band model to ensure the connectivity of its GBZ. Breaking these conditions allows us to explicitly construct disconnected GBZs, with more connected components than the number of bands in multiband models. This novel GBZ topology is applied to further demonstrate a counterintuitive effect, where the line gap of a two-band open-boundary spectrum with sublattice symmetry may be closed without changing its point-gap topology. Our results challenge the current understanding of bands and gaps in non-Hermitian systems and highlight the need to further investigate the topological effects associated with the GBZ, including topological invariants and open-boundary braiding.