Instability of the ferromagnetic quantum critical point and symmetry of the ferromagnetic ground state in two-dimensional and three-dimensional electron gases with arbitrary spin-orbit splitting

It is well known that in the absence of the spin-orbit (SO) splitting the zero-temperature ferromagnetic phase transition in two-dimensional (2D) and three-dimensional (3D) electron gas is discontinuous (first order). The physical reason for this effect lies in the infrared catastrophe brought by the long-range particle-hole fluctuations near the Fermi surface. It is widely believed that a finite SO splitting is able to regularize this infrared catastrophe, and therefore, to stabilize the ferromagnetic quantum critical point. In contrast to this, we show that the infrared catastrophe persists at arbitrary SO splitting and the zero-temperature ferromagnetic phase transition in the itinerant 2D and 3D electron gas is always discontinuous. We also find that SO splitting reduces the symmetry of the ferromagnetic ground state down to the symmetry of the spin-orbit term. For example, Rashba SO splitting in 2D electron gas leads to the easy-plane symmetry of the ferromagnetic ground state. A combination of the Rashba SO splitting with the Dresselhaus term reduces the symmetry of the ferromagnetic ground state down to the in-plane Ising ferromagnet. The infrared catastrophe can be measured via the nonanalytic dependence of the spin susceptibility on magnetic field. This dependence is strongly anisotropic and follows the symmetry of SO splitting.