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 χij on the momentum (q) and on the SOI coupling constant (α). The uniform (q=0) spin susceptibility is anisotropic (with the out-of-plane component χzz being larger than the in-plane one χxx by a term proportional to U2(2k _F)|α|, where U(q) is the electron-electron interaction). For q≤2m ^*|α|, corrections to the leading U2(2k _F)|α| term scale linearly with q for χxx and are absent for χzz. 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=0. To second order in U(q), the dispersion is a decreasing function of q, and anisotropy is not sufficient to stabilize long-range order. However, renormalization in the Cooper channel for q2m ^*|α| is capable of reversing the sign of the q dependence of χxx 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 π/m ^*|α|.