Selected for a Viewpoint in Physics

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