Robustness of unconventional $s$-wave superconducting states against disorder

We investigate the robustness against disorder of superconductivity in multiband systems where the fermions have four internal degrees of freedom. This permits unconventional $s$-wave pairing states, which may transform nontrivially under crystal symmetries. Using the self-consistent Born approximation, we develop a general theory for the effect of impurities on the critical temperature, and find that the presence of these novel $s$-wave channels significantly modifies the conclusions of single-band theories. We apply our theory to two candidate topological superconductors, YPtBi and Cu$_x$Bi$_2$Se$_3$, and show that the novel $s$-wave states display an enhanced resilience against disorder, which extends to momentum-dependent pairing states with the same crystal symmetry. The robustness of the $s$-wave states can be quantified in terms of their superconducting fitness, which can be readily evaluated for model systems.