Fermion doubling theorems in 2D non-Hermitian systems for Fermi points and exceptional points

TL;DR

This paper extends the fermion doubling theorem to two-dimensional non-Hermitian systems, demonstrating that both Fermi points and exceptional points must come in pairs, and introduces a new invariant called the discriminant number.

Contribution

It introduces a doubling theorem for non-Hermitian 2D systems and the discriminant number invariant applicable to exceptional points of any order.

Findings

Fermi points and exceptional points obey doubling theorems in 2D non-Hermitian systems

The discriminant number invariant characterizes exceptional points and degeneracies

Surface states can violate the doubling theorem, indicating unusual bulk physics

Abstract

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the discriminant number. Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order, and moreover can also be used to characterize non-defective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.