Generalized Brillouin Zone Fragmentation

Abstract

The Generalized Brillouin Zone (GBZ) encodes how lattice momentum is complex-deformed due to non-Hermitian skin accumulation, and has proved essential in restoring bulk-boundary correspondences. However, we find that generically, the GBZ is neither unique nor well-defined if more than one skin localization direction or strength exists, even in systems with no asymmetric hoppings. Instead, open boundary condition (OBC) eigenstates become complicated superpositions of multiple competing skin modes from "fragments" of all possible GBZs solutions. We develop a formalism that computes the fragmented GBZ in a scalable manner, with fragmentation extent quantified through our newly-defined composition IPR and spectral relative entropy. GBZ fragmentation is revealed to fundamentally challenge the notion of discontinuous phase transitions, since topological winding contributions from different GBZ fragments can "melt away" at different rates. Phenomenologically, GBZ fragmentation also leads to edge localization in all observables in energetically weighted ensembles such as thermal ensembles. This contrasts with conventional GBZs where the skin localization completely cancels in biorthogonal expectations. Occurring universally in multi-mode non-Hermitian media, as we concretely demonstrate with photonic crystal simulations, GBZ fragmentation points towards a new paradigm that is essential for understanding the band structure and the topological and dynamical properties of diverse generic non-Hermitian systems.