Spin responses of a disordered helical superconducting edge under a Zeeman field

We investigate analytically and numerically the effects of disorder on the helical edge of the 2D topological insulator in the presence of the Zeeman field and superconductivity. Employing bosonization and a renormalization-group analysis, we study how impurity potentials modify charge- and spin-density wave correlations as well as superconducting pair correlations. Our results reveal that the Zeeman field controls the competition: in the attractive regime, it amplifies the superconducting gap, while in the repulsive regime, it stabilizes impurity effects by keeping the system longer in the relevant regime for disorder. We also find that disorder induces logarithmic suppression of transverse density-wave correlations, while at the same time introducing positive logarithmic corrections that enhance superconducting pair correlations and contribute to their stability. These effects directly modify the scaling of spin conductance, providing experimentally accessible signatures of the interplay between disorder and superconductivity in topological edge channels.