Electronic liquid crystalline phases in a spin-orbit coupled two-dimensional electron gas

We argue that the ground state of a two-dimensional electron gas with Rashba spin-orbit coupling realizes one of several possible liquid crystalline or Wigner crystalline phases in the low-density limit, even for short-range repulsive electron-electron interactions (which decay with distance with a power larger than two). Depending on specifics of the interactions, preferred ground states include an anisotropic Wigner crystal with an increasingly anisotropic unit cell as the density decreases, a striped or electron smectic phase, and a ferromagnetic phase that strongly breaks the lattice point-group symmetry, i.e., exhibits nematic order. Melting of the anisotropic Wigner crystal or the smectic phase by thermal or quantum fluctuations can likely give rise to a nonmagnetic nematic phase that preserves time-reversal symmetry.

Figure 1

(a) Dispersion of a particle with Rashba SOC. The minimum of the dispersion occurs on a ring in $k$ space, marked by a red circle. The red arrows show the spin polarization of the different Bloch states. (b) Density of states as a function of energy, corresponding to the dispersion shown in (a). Near the band bottom, the density of states diverges as $\rho \left(E\right)\sim {\left(E\right)}^{-1/2}$. (c) Schematic phase diagram in the low-density limit with repulsive electron-electron interactions that decay at long distances as $V\sim r^{-\alpha }$. For $\alpha ⩽2$, the ground state is an isotropic Wigner crystal (i.e., it preserves a discrete rotational symmetry, C${}_n$ with $n>2$). For $\alpha >2$, states with a further broken rotational symmetry are favored: (i) represents an anisotropic Wigner crystal with a unit cell, which becomes parametrically anisotropic in the low-density limit, (ii) and (iii) represent snapshots of a smectic state and a ferromagnetic nematic liquid state, respectively.

Figure 2

The quantities $S_{x,y,z}$, defined in Eq. (20) vs the aspect ratio $\eta =L_x/L_y$ of a box with fixed area $\Omega =75/k_0^2$. When $L_x/L_y=1$, $S_x=S_y$; as $L_x/L_y$ increases, $S_y$ becomes close to 1 and $S_x\to 0$.

Figure 3

Sketch of the phase diagram of a 2DEG with Rashba SOC, as a function of $1/r_s\equiv a_0{\left(\pi n\right)}^{1/2}$ and the screening length $\xi$, where $a_0=\kappa /m{e}^2$ is the Bohr radius. The three regions correspond to the uniform Fermi liquid (UFL), the isotropic Wigner crystal (WC), and a phase featuring a high degree of anisotropy, which can be either an anisotropic Wigner crystal, a smectic, or a nematic. The red dashed lines correspond to $r_s\sim \xi /a_0$, where the crossover between effectively short-range (screened) and long-range Coulomb interactions occurs; $1/r_s\sim k_0{a}_0$, where the SOC length scale becomes comparable to the interparticle distance; and $1/r_s\sim {(\xi /a_0)}^{-1/2}$, where the energies of the uniform Fermi liquid and the anisotropic phases become comparable, see Eq. (25).

Figure 4

Dispersion of the lowest subband for infinite strips of varying width $L_y$. The inset shows the dispersion minimum $\Delta$ vs $L_y$ on a log-log plot, showing good agreement with $\Delta \sim 1/L_y^4$.

Figure 5

Numerical results for the ground-state energy vs aspect ratio $L_x/L_y$ of a particle with Rashba spin-orbit coupling in a rectangular infinite potential well of size $L_x\times L_y$, for several values of the density $n$ (measured in units of $1/k_0^2$). We have set $2m=1$.

Figure 6

(a) Minimal ground-state energy, ${\varepsilon }^{☆}$, and (b) optimal aspect ratio, ${\eta }^{☆}$, as a function of the density, on a log-log scale. The data are consistent with the analytical predictions: ${\varepsilon }^{☆}\sim n^{4/3}$ and ${\eta }^{☆}\sim n^{-1/3}$ at low densities.

Figure 7

Constant-energy contours of the dispersion of the mean-field Hamiltonian, Eq. (B2), around the minimum at $\vec{k}=k_0\widehat{\mathbf{y}}$. Numerical labels indicate energy values above the minimum, measured in units of $k_0^2/\left(2m\right)$. The dashed line indicates the circle $k_x^2+k_y^2=k_0^2$.