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The quantum correction to the conductivity in disordered quantum wires withlinear Rashba spin-orbit coupling is obtained. For quantum wires withspin-conserving boundary conditions, we find a crossover from weak anti- toweak localization as the wire width W is reduced using exact diagonalization ofthe Cooperon equation. This crossover is due to the dimensional dependence ofthe spin relaxation rate of conduction electrons, which becomes diminished,when the wire width is smaller than the bulk spin precession length $L_{SO}$.We thus confirm previous results for small wire width, $W/L_{SO}<= 1$[PRL98,176808(2007)], where only the transverse 0-modes of the Cooperonequation had been taken into account. We find that spin helix solutions becomestable for arbitrary ratios of linear Rashba and Dresselhaus coupling in narrowwires. For wider wires, the spin relaxation rate is found to be not monotonousas function of wire width: it becomes first enhanced for W on the order of thebulk $L_{SO}$ before it becomes diminished for smaller wire widths. Inaddition, we find that the spin relaxation is smallest at the edge of the wirefor wide wires. The effect of the Zeeman coupling to the magnetic fieldperpendicular to the 2D electron system is studied and found that it shifts thecrossover from weak anti- to weak localization to larger wire widths $W_c$.When the transverse confinement potential of the quantum wire is smooth(adiabatic), the spin relaxation rate is found to be enhanced as W is reduced.We find that only a spin polarized state retains a finite spin relaxation ratein such narrow wires. Thus, we conclude that the injection of polarized spinsinto nonmagnetic quantum wires should be favorable in wires with smoothconfinement potential. Finally, in wires with tubular shape, corresponding totransverse periodic boundary conditions, we find no reduction of the spinrelaxation rate.

Dimensional Dependence of Weak Localization Corrections and Spin Relaxation in Quantum Wires with Rashba Spin-Orbit Coupling