  Recently, topological quantum states of non-Hermitian systems, exhibiting
rich new exotic states, have attracted great attention in condensed-matter
physics. As for the demonstration, most of non-Hermitian topological phenomena
previously focused on are in one- and two-dimensional systems. Here, we
investigate three-dimensional non-Hermitian nodal-line semimetals in the
presence of a particle gain-and-loss perturbation. It is found that this
perturbation will split the original nodal ring into two exceptional rings
(ERs). The topological nature of the bulk electronic structure is characterized
by two different topological invariants, namely, the vorticity and the winding
number defined for a one-dimensional loop in momentum space, both of which are
shown to take half-integer (integer) values when an odd (even) number of ERs
thread through the loop. The conventional bulk-surface correspondence in
non-Hermitian nodal-line semimetals is found to break down, where the surface
zero-energy flat bands are no longer bounded by projections of bulk ERs.
Alternatively, a macroscopic fraction of the bulk eigenstates can be localized
near the surface, thus leading to the so-called non-Hermitian skin effect.
