Superconductivity in disordered locally noncentrosymmetric materials: an application to CeRh$_2$As$_2$

Layered three-dimensional centrosymmetric crystals can exhibit characteristics of noncentrosymmetric materials. This happens when each individual layer alone lacks inversion, but, when combined, inversion symmetry is restored; hence the designation: locally noncentrosymmertic superconductors (LNCSs). In LNCSs, the effects of impurities and subdominant magnetic field induced pairing channels remain unexplored. Using a minimal model, we examine all pairing channels and show that there is always a subdominant superconducting instability that is favored at high magnetic fields, which can substantially alter the magnetic field -- temperature phase diagram. Also, we find that the phase diagram responds to disorder in a non-monotonic way, which can be subjected to experimental verification. We apply these ideas to the recently unveiled two-phase superconducting phase diagram of CeRh$_2$As$_2$. We identify the two phases as singlet-triplet mixed even- and odd-parity states at low and a high fields, respectively. Furthermore, we predict the presence of two superconducting phases also for in-plane magnetic fields in cleaner samples, since a high-field phase could have been so far hindered by impurity effects.