Non-Hermitian spatial symmetries and their stabilized normal and exceptional topological semimetals

We study non-Hermitian spatial symmetries -- a class of symmetries that have no counterparts in Hermitian systems -- and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. In deriving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.