Competing Effects of Interactions and Spin-Orbit Coupling in a Quantum Wire

We study the interplay of electron-electron interactions and Rashba spin-orbit coupling in one-dimensional ballistic wires. Using the renormalization group approach we construct the phase diagram in terms of Rashba coupling, Tomonaga-Luttinger stiffness and backward scattering strength. We identify the parameter regimes with a dynamically generated spin gap and show where the Luttinger liquid prevails. We also discuss the consequences for the operation of the Datta-Das transistor.

Figure 1

Flow diagram for the ($y_R$,$\frac{1}{2}\ln K_s$) plane for $y_{R,0}=0.07$ and various initial values $K_0$. Full lines correspond to $y_{\perp ,0}=0$, dashed lines to $y_{\perp ,0}=0.4$.

Figure 2

Phase diagram in the parameter space of initial values $y_{R,0}$ and $\frac{1}{2}\ln (K_0)$ for $y_{\perp ,0}=0$ (full lines), $y_{\perp ,0}=0.4$ (dashed lines), and for $y_{f,0}=0$. The figure exhibits clearly the dual symmetry of Eq. (5). Spin gaps exist both in the phase with dominant superconducting correlations (SS or ${\mathrm{TS}}_{(x)}$) and in the spin (charge)-density wave phases (${\mathrm{SDW}}_{(x)}$ or CDW). In the Luttinger-liquid phase (${\mathrm{TL}}_{(\mathrm{inc})}$) all correlation functions decay algebraically. The TL phase is separated from the two spin-gapped phases by commensurate-incommensurate transitions (C-IC lines). The bold dotted line (AT at $K_s=1$) corresponds to the critical Ashkin-Teller model.