One-dimensional non-Hermitian band structures as Riemann surfaces

Abstract

Non-Hermitian periodic systems possess unique properties not found in their Hermitian counterparts, including non-Hermitian skin effects in the open-boundary-condition spectrum and nontrivial braiding of the bulk band structure. Here, by viewing one-dimensional non-Hermitian band structures as Riemann surfaces, we show that the monodromy representation, a group homomorphism from the fundamental group of a punctured complex plane to the permutation group operating on the ordering of the Riemann sheets, serves as a topological invariant of the system. The connection between monodromy representations and the experimental observable effects is established through the branch cuts induced by the underlying multivalued functions. An open-boundary spectrum is interpreted as branch cuts connecting certain branch points, and its consistency with the monodromy representation severely limits its possible morphology. A braid word along a closed loop is controlled by the number and permutation labels of branch points within the loop, and its crossing number is given by the winding number of the discriminant. The importance of the monodromy representation as a topological invariant and the analysis of the Riemann surface geometry as defined by the band structure can be used to generate important insights about the physical behaviors of non-Hermitian systems.