Phase diagram of dirty two-band superconductors and observability of impurity-induced s+is state

We investigate the phase diagram of dirty two-band superconductors. This paper primarily focuses on the properties and observability of the time-reversal symmetry-breaking $s+is$ superconducting states, which can be generated in two-band superconductors by interband impurity scattering. We show that such states can appear in two distinct ways. First, according to a previously discussed scenario, the $s+is$ state can form as an intermediate phase at the impurity-driven crossover between $s_{\pm}$ and $s_{++}$ states. We show that there is a second scenario where domains of the $s+is$ state exists in the form of an isolated dome inside the $s_{\pm}$ domain, completely detached from the transition between $s_{\pm}$ and $s_{++}$ states. We demonstrate that in both cases the $s+is$ state, generated by impurity scattering exists in an extremely small interval of impurity concentrations. Although this likely precludes direct experimental observation of the $s+is$ state formation due to this mechanism, this physics leads to the appearance of a region inside both the $s_{\pm}$ and $s_{++}$ domains with unusual properties due to softening of normal modes.