Generalized Stoner criterion and versatile spin ordering in two-dimensional spin-orbit coupled electron systems

Spin-orbit coupling is a single-particle phenomenon known to generate topological order, and electron-electron interactions cause ordered many-body phases to exist. The rich interplay of these two mechanisms is present in a broad range of materials and has been the subject of considerable ongoing research and controversy. Here we demonstrate that interacting two-dimensional electron systems with strong spin-orbit coupling exhibit a variety of time reversal symmetry breaking phases with unconventional spin alignment. We first prove that a Stoner-type criterion can be formulated for the spin polarization response to an electric field, which predicts that the spin polarization susceptibility diverges at a certain value of the electron-electron interaction strength. The divergence indicates the possibility of unconventional ferromagnetic phases even in the absence of any applied electric or magnetic field. This leads us, in the second part of this work, to study interacting Rashba spin-orbit coupled semiconductors in equilibrium in the Hartree-Fock approximation as a generic minimal model. Using classical Monte Carlo simulations, we construct the complete phase diagram of the system as a function of density and spin-orbit coupling strength. It includes both an out-of-plane spin-polarized phase and in-plane spin-polarized phases with shifted Fermi surfaces and rich spin textures, reminiscent of the Pomeranchuk instability, as well as two different Fermi-liquid phases having one and two Fermi surfaces, respectively, which are separated by a Lifshitz transition. We discuss possibilities for experimental observation and useful application of these novel phases, especially in the context of electric-field-controlled macroscopic spin polarizations.

Figure 1

Phase diagram of 2D electron liquid with Rashba spin-orbit coupling obtained by solving the Hartree-Fock equations using a Monte Carlo method. Here, $\tilde{\alpha }$ and $r_s$ are dimensionless measures for the strength of Rashba spin-orbit coupling and electron-electron interactions, respectively: $\tilde{\alpha }$ corresponds to the ratio of Fermi wavelength and spin-precession length, and $r_s$ is the Wigner-Seitz radius of the 2D electron system. The distinguishing ground-state features for each individual phase are indicated schematically. For the paramagnetic Fermi-liquid phase FL1 (FL2), there is no net spin polarization, and the ground state is a Fermi sea formed from one (both) spin subband(s). In contrast, the OP phase is characterized by a centered Fermi surface and an out-of-plane magnetization. The IP phase is the most unconventional, exhibiting an in-plane magnetization associated with a shifted Fermi sea.

Figure 2

The total energy $E_{\text{tot}}$ plots of the FL2 (black solid line) and OP (orange dashed line) states vs $r_s$ at $\tilde{\alpha }=0.12$. The units of the energy are $N_e{\hslash }^2{k}_{\mathrm{F}}^2/2m$, where $N_e=nA$ is the total electron number. We can see the linear $r_s$ dependence of the total energies when $r_s$ is close to phase boundaries, which allows us to use the linear fitting to determine transition points.

Figure 3

The OP phase at $\tilde{\alpha }=0.3$ and $r_s=2.02$ in a 3D view. The OP phase comprises of a single band with circular Fermi surface and nontrivial out-of-plane spin polarization. Its in-plane spin polarization cancels out after summing over all the occupied states.

Figure 4

(a) and (b) The IP phase at $\tilde{\alpha }=0.12$ and $r_s=2.25$, and $\tilde{\alpha }=1.13$ and $r_s=2.12$. In (a), the ratio $\tilde{\alpha }/r_s\sim 0.1$ is small and the Fermi surface is roughly symmetric with spins almost parallel aligned. In (b), the ratio $\tilde{\alpha }/r_s\sim 1$ is large and the Fermi surface is deformed into a “heart” shape. The blue dot in (a) indicates the center of the displaced Fermi surface. The spin texture in (b) follows the Fermi surface deformation, and the spins are winding around a center below the $x$ axis.

Figure 5

Solution of Eq. (46). The function $\delta \theta \left(p\right)$ determines the spin texture of the ferromagnetic states at small $\alpha$, see Eq. (47), as well as the critical point $r_s^{**}$, see Eq. (54).