Development and current situation of generalized Brillouin zone theory

The non-Hermitian skin effect is a unique phenomenon in non-Hermitian systems, characterized by the localization of a large number of eigenstates at the system’s boundaries. While the one-dimensional generalized Brillouin zone (GBZ) theory has provided a comprehensive understanding of this effect in one-dimensional systems, the establishment of a high-dimensional GBZ theory still faces significant challenges due to difficulties in numerical and analytical calculations. Addressing issues such as energy spectra, density of states, localized eigen function patterns, and the influence of open boundary geometries, this paper reviews and summarizes the current developments in two-dimensional GBZ theory. The discussion primarily encompasses the following five aspects: (1) Building on the well-established one-dimensional GBZ theory, we identify and propose critical scientific questions for the two-dimensional GBZ theory; (2) introducing the Amoeba theory, which utilizes Ronkin functions to analyze the structure of the GBZ; (3) drawing an analogy to two-dimensional electrostatics, this theory is constructed to describe the energy spectrum; (4) developing a dynamical duality method to construct eigen functions; (5) advancing the generalized Fermi surface theory based on transfer matrix techniques to explore high-dimensional systems.