Ferromagnetic order of nuclear spins coupled to conduction electrons : a combined effect of electron-electron and spin-orbit interactions

We analyze the ordered state of nuclear spins embedded in an interacting two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI). Stability of the ferromagnetic nuclear-spin phase is governed by nonanalytic dependences of the electron spin susceptibility chi(ij) on the momentum ((q) over tilde) and on the SOI coupling constant (alpha). The uniform ((q) over tilde = 0) spin susceptibility is anisotropic (with the out-of-plane component chi(zz) being larger than the in-plane one chi(xx) by a term proportional to U-2(2k(F))vertical bar alpha vertical bar, where U(q) is the electron-electron interaction). For (q) over tilde >= 2m*vertical bar alpha vertical bar, corrections to the leading U-2(2kF)vertical bar alpha vertical bar term scale linearly with (q) over tilde for chi(xx) and are absent for chi(zz). This anisotropy has important consequences for the ferromagnetic nuclear-spin phase: (i) the ordered state-if achieved-is of an Ising type and (ii) the spin-wave dispersion is gapped at (q) over tilde = 0. To second order in U(q), the dispersion is a decreasing function of (q) over tilde, and anisotropy is not sufficient to stabilize long-range order. However, renormalization in the Cooper channel for (q) over tilde >> 2m*vertical bar alpha vertical bar is capable of reversing the sign of the (q) over tilde dependence of chi(xx) and thus stabilizing the ordered state. We also show that a combination of the electron-electron and spin-orbit (SO) interactions leads to a new effect: long-wavelength Friedel oscillations in the spin (but not charge) electron density induced by local magnetic moments. The period of these oscillations is given by the SO length pi/m*vertical bar alpha vertical bar.