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The two-dimensional electron gas is of fundamental importance in quantum many-body physics. We study a minimal extension of this model with C₄ (as opposed to full rotational) symmetry and an electronic dispersion with two valleys with anisotropic effective masses. Electrons in our model interact via Coulomb repulsion, screened by distant metallic gates. Using variational Monte Carlo simulations, we find a broad intermediate range of densities with a metallic valley-polarized, spin-unpolarized ground state. Our results are of direct relevance to the recently discovered ``nematic'' state in AlAs quantum wells. For the effective mass anisotropy relevant to this system, mₗ/mₘ5. 2, we obtain a transition from an anisotropic metal to a valley-polarized metal at rₒ12 (where rₒ is the dimensionless Wigner-Seitz radius). At still lower densities, we find a (possibly metastable) valley and spin-polarized state with a reduced electronic anisotropy.
Valenti et al. (Mon,) studied this question.