We describe a case of instability of the paramagnetic order in an electron gas confined in a thin tubular semiconductor nanowire with a Rashba spin-orbit interaction. This situation is realized when, for a critical value of the Rashba coupling constant ${\alpha }_c$, dependent only on the radius of the tube, single-particle energies corresponding to opposite spin orientations along the direction tangent to the tube become equal. A theory of the magnetic phase transition considers the presence of the Coulomb interaction, evaluated within the Hartree-Fock approximation, which lifts the spin degeneracy and leads to the formation of quasiparticle states that support a continuous rotation of the electron spin in the momentum space along the axis of the wire. For a chemical potential localized within the gap between the quasiparticle states, the long-range magnetic order manifests macroscopically through a constant radial spin polarization proportional with the magnitude of the gap.

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• Spin density waves
• Spin-orbit coupling
• Spintronics
• Nanowires
• Hartree-Fock methods