  Open quantum systems display unusual phenomena not seen in closed systems,
such as new topological phases and unconventional phase transitions. An
interesting example was studied for a quantum spin liquid in the Kitaev model
[K. Yang, S. C. Morampudi, and E. J. Bergholtz, Phys. Rev. Lett. ${\bf 126}$,
077201 (2021)]; an effective non-Hermitian Kitaev model, which incorporates
dissipation effects, was shown to give rise to a gapless spin liquid state with
exceptional points in the Majorana dispersions. Given that an external magnetic
field induces a gapped Majorana topological state in the Hermitian case, the
exceptional points may bring about intriguing quantum phenomena under a
magnetic field. Here we investigate the non-Hermitian Kitaev model perturbed by
the magnetic field. We show that the exceptional points remain gapless up to a
finite critical magnetic field, in stark contrast to the Hermitian case where
an infinitesimal field opens a gap. The gapless state is stable over a wide
range of the magnetic field for some particular parameter sets, and in special
cases, undergoes topological transitions to another gapless state with
different winding number around the exceptional points without opening a gap.
In addition, in the system with edges, we find that the non-Hermitian skin
effect is induced by the magnetic field, even for the parameters where the skin
effect is absent at zero field. The chirality of edge states is switched
through the exceptional points, similarly to the surface Fermi arcs connected
by the Weyl points in three-dimensional Weyl semimetals. Our results provide a
new possible route to stabilize topological gapless quantum spin liquids under
the magnetic field in the presence of dissipation.
