We study theoretically the role of electron-electron interactions in one-dimensional magnetized helical states coupled to an $s$-wave superconductor. We consider a partially mixed helical (superhelical) regime, where the magnetic field (superconductivity) has a dominant contribution. Using bosonization and renormalization-group techniques, it is shown how the interactions affect the correlation functions in the system. In the partially mixed helical regime, we find that superconducting pairing promotes spin-density-wave correlations, while singlet and triplet pairing correlations are suppressed, especially under attractive interactions. In contrast, in the superhelical regime, a perturbative Zeeman field enhances both spin singlet and spin triplet-$x$ pairings under repulsive interactions. We calculate both logarithmic and residual corrections to charge-density-wave, spin-density-wave, and pairing correlations, revealing short- and long-range behaviors. We further investigate spin transport properties supplemented by a renormalization-group analysis of the temperature and frequency dependence of spin conductivity. We also analyze the transport of momentum-spin-locked carriers in the presence of Zeeman-induced and pairing-induced gaps, uncovering the effect of interactions and the interplay between the two gaps in helical systems.

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• Edge states
• Magnetism
• Superconductivity
• 1-dimensional systems
• Luttinger liquid model