  Due to the fundamental position of spin-orbit coupled ultracold atoms in the
simulation of topological insulators, the gain/loss effects on these systems
should be evaluated when considering the measurement or the coupling to the
environment. Here, incorporating the mature gain/loss techniques into the
experimentally realized spin-orbit coupled ultracold atoms in two-dimensional
optical lattices, we investigate the corresponding non-Hermitian tight-binding
model and evaluate the gain/loss effects on various properties of the system,
revealing the interplay of the non-Hermiticity and the spin-orbit coupling.
Under periodic boundary conditions, we analytically obtain the topological
phase diagram, which undergoes a non-Hermitian gapless interval instead of a
point that the Hermitian counterpart encounters for a topological phase
transition. We also unveil that the band inversion is just a necessary but not
sufficient condition for a topological phase in two-level spin-orbit coupled
non-Hermitian systems. Because the nodal loops of the upper or lower two
dressed bands of the Hermitian counterpart can be split into exceptional loops
in this non-Hermitian model, a gauge-independent Wilson-loop method is
developed for numerically calculating the Chern number of multiple degenerate
complex bands. Under open boundary conditions, we find that the conventional
bulk-boundary correspondence does not break down with only on-site gain/loss
due to the lack of non-Hermitian skin effect, but the dissipation of chiral
edge states depends on the boundary selection, which may be used in the control
of edge-state dynamics. Given the technical accessibility of state-dependent
atom loss, this model could be realized in current cold-atom experiments.
