Variation of the elliptical Fermi surface for a two-dimensional electron gas with anisotropic mass

We consider a two-dimensional electron gas in the thermodynamic (bulk) limit. It is assumed that the system consists of fully spin-polarized (spinless) electrons with anisotropic mass. We study the variation of the shape of the expected elliptical Fermi surface as a function of the density of the system in presence of such form of internal anisotropy. To this effect, we calculate the energy of the system as well as the optimum ellipticity of the Fermi surface for two possible liquid states. One corresponds to the standard system with circular Fermi surface while the second one represents a liquid anisotropic phase with a tunable elliptical deformation of the Fermi surface that includes the state that minimizes the kinetic energy. The results obtained shed light on several possible scenarios that may arise in such a system. The competition between opposing tendencies of the kinetic energy and potential energy may lead to the stabilization of liquid phases where the optimal elliptical deformation of the Fermi surface is non-obvious and depends on the density as well as an array of other factors related to the specific values of various parameters that characterize the system