Bogoliubov Fermi surfaces due to out-of-plane magnetic fields in monolayer superconducting NbSe$_2$*

We use a low-energy model to investigate the effect of an out-of-plane magnetic field on the superconducting properties of monolayer NbSe$_2$.  We find that, even when the singlet-channel interactions dominate, such that the zero-field superconducting state is fully gapped, there is a threshold Zeeman field beyond which the quasiparticle spectrum is no longer fully gapped, giving rise to Bogoliubov Fermi surfaces.  We establish this by self-consistently solving the gap equations for mixed singlet and triplet pairing on the $\Gamma$ pocket, in the presence of both Ising spin-orbit coupling and Zeeman field.  We find that the resulting evolution of the density of states with magnetic field qualitatively agrees with the scanning tunneling microscopy data in this material.