Josephson junction of nodal superconductors with a Rashba and Ising spin-orbit coupling

We study the effect of a Rashba spin-orbit coupling on the nodal superconducting phase of an Ising superconductor. Such nodal phase was predicted to occur when applying an in-plane field beyond the Pauli limit to a superconducting monolayer transition metal dichalcogenides (TMD). Generically, Rashba spin orbit is known to lift the chiral symmetry that protects the nodal points, resulting in a fully gapped phase. However, when the magnetic field is applied along the $x$ direction, a residual vertical mirror symmetry protects a nodal crystalline phase. We study a single-band tight-binding model that captures the low-energy physics around the $\mathrm{\ensuremath{\Gamma}}$ pocket of monolayer TMD. We calculate the topological properties, the edge state structure, and the current phase relation in a Josephson junction geometry of the nodal crystalline phase. We show that while the nodal crystalline phase is characterized by localized edge modes on non-self-reflecting boundaries, the current phase relation exhibits a trivial $2\ensuremath{\pi}$ periodicity in the presence of Rashba spin-orbit coupling.