  Non-Hermitian systems have been discussed mostly in the context of open
systems and nonequilibrium. Recent experimental progress is much from optical,
cold-atomic, and classical platforms due to the vast tunability and clear
identification of observables. However, their counterpart in solid-state
electronic systems in equilibrium remains unmasked although highly desired,
where a variety of materials are available, calculations are solidly founded,
and accurate spectroscopic techniques can be applied. We demonstrate that, in
the surface state of a topological insulator with spin-dependent relaxation due
to magnetic impurities, highly nontrivial topological soliton spin textures
appear in momentum space. Such spin-channel phenomena are delicately related to
the type of non-Hermiticity and correctly reveal the most robust non-Hermitian
features detectable spectroscopically. Moreover, the distinct topological
soliton objects can be deformed to each other, mediated by topological
transitions driven by tuning across a critical direction of doped magnetism.
These results not only open a solid-state avenue to exotic spin patterns via
spin- and angle-resolved photoemission spectroscopy, but also inspire
non-Hermitian dissipation engineering of spins in solids.
