Deformation of the Fermi surface of a spinless two-dimensional electron gas in presence of an anisotropic Coulomb interaction potential

We consider the stability of the circular Fermi surface of a two-dimensional electron gas system against an elliptical deformation induced by an anisotropic Coulomb interaction potential. We use the jellium approximation for the neutralizing background and treat the electrons as fully spin-polarized (spinless) particles with a constant isotropic (effective) mass. The anisotropic Coulomb interaction potential considered in this work is inspired from studies of two-dimensional electron gas systems in the quantum Hall regime. We use a Hartree–Fock procedure to obtain analytical results for two special Fermi liquid quantum electronic phases. The first one corresponds to a system with circular Fermi surface while the second one corresponds to a liquid anisotropic phase with a specific elliptical deformation of the Fermi surface that gives rise to the lowest possible potential energy of the system. The results obtained suggest that, for the most general situations, neither of these two Fermi liquid phases represent the lowest energy state of the system within the framework of the family of states considered in this work. The lowest energy phase is one with an optimal elliptical deformation whose specific value is determined by a complex interplay of many factors including the density of the system.

where D k is the region bounded by an ellipse in k-space. The integral above is simple and the final result is written as: where we used the fact that the total number of spinless electrons is: Based on the definition of parameter α from Eq.(12), one writes the kinetic energy per particle as: is the Fermi energy for a circular Fermi surface.

APPENDIX B: ONE-PARTICLE DENSITY MATRIX
By transforming the sum over k into an integral over k one can write the one-particle density matrix in the thermodynamic limit as: where α is defined in Eq.(12) and D k is the region bounded by the ellipse in k-space. The integration can be done analytically and the final result is: where ρ 0 = k a k b /(4 π) = k 2 F /(4 π) is the uniform density of the system, J 1 (x) is a Bessel function of the first kind and r 21 = r 2 − r 1 = (x 21 , y 21 ) is a separation vector. One has α = 1 for a circular Fermi surface which leads to the following expression for the one-particle density matrix: where r 21 = x 2 21 + y 2 21 is the radial separation distance.

APPENDIX C: POTENTIAL ENERGY
In order to calculate the total potential energy in Eq.(15) we use the expression for ρ(α, r 1 , r 2 ) given from Eq.(B2) and write the total potential energy as: where The potential energy per particle in the thermodynamic limit can be written as: In the thermodynamic limit, L → ∞ which means that −∞ < x 21 , y 21 < +∞. Therefore, the quantity to calculate is: where we substituted the form of v γ (x 21 , y 21 ) from Eq.(C2). We now introduce new auxiliary variables that make the one-particle density matrix "isotropic", namely, we introduce the following new variables defined as: With this transformation the quantity in Eq.(C4) can be rewritten as: Now we change to a 2D polar system of coordinates: x = r cos(ϕ ) ; y = r sin(ϕ ) .
This additional change of variables allows us to write the potential energy per particle as: .

(C8)
The integration over the radial variable can be done with the following integral formula: The integration over the angular variable can be done with the following integral formula: 2π 0 dϕ a 2 cos 2 (ϕ) + b 2 sin 2 (ϕ) where K(m) is a complete elliptic integral of the first kind with parameter, m defined as By using the formula in Eq.(C10) it is easy to prove that: Let's define an auxiliary function F (x) written as: After some simple algebraic manipulations and relying on the fact that ρ 0 = k 2 F /(4π) we can write the potential energy per particle as: u(α, γ) = − 8 3π 2 k F k e e 2 F (α γ) , where F (x) is the auxiliary function in Eq.(C13). Note that the argument of F (x) in Eq.(C14) is the product of parameters α and γ.