  For first-order topological semimetals, non-Hermitian perturbation can drive
the Weyl nodes into Weyl exceptional rings having multiple topological
structures and no Hermitian counterparts. Recently, it was discovered that
higher-order Weyl semimetals, as a novel class of higher-order topological
phases, can uniquely exhibit coexisting surface and hinge Fermi arcs. However,
non-Hermitian higher-order topological semimetals have not yet been explored.
Here, we identify a new type of topological semimetals, i.e, a higher-order
topological semimetal with Weyl exceptional rings. In such a semimetal, these
rings are characterized by both a spectral winding number and a Chern number.
Moreover, the higher-order Weyl-exceptional-ring semimetal supports both
surface and hinge Fermi-arc states, which are bounded by the projection of the
Weyl exceptional rings onto the surface and hinge, respectively. Our studies
open new avenues for exploring novel higher-order topological semimetals in
non-Hermitian systems.
