Stability of Kramers Majorana doublets: the effects of interactions and disorders

In this work we study the effects of interactions and disorder on $1D$ DIII topological superconductors and the Majorana Kramers doublets (MKDs). In contract to the case without the time-reversal symmetry, the Umklapp interaction plays important roles in this system. The underlying phases due to the Umklapp interaction and disorder are found by using a perturbative renormaliztion analysis based on the Abelian Bosonization. Importantly, the stable topological regime can be found within a rather wide parameter space. Furthermore, the degeneracy splitting of the MKDs is shown to be still exponentially dependent on the length of the wire in the presence of both the Umklapp interaction and disorder, when the Luttinger parameter $K_0>\sqrt{2}/2$. The differences caused by the Umklapp interaction are highlighted in contrast to the time-reversal breaking cases.