Charge density waves and superconductivity in the electron-positive fermion gas using a simple intuitive model. Part I: The model, instabilities, and phase diagram

Abstract

The electron-positive fermion gas in three dimensions and $T=0$ is modeled as two independent fermion gases interacting via the coulomb interaction. The main advantage of the simple model is that all existing results from the electron gas can be directly used for the positive fermion gas, which is the same as the electron gas, but scaled for the mass of the positive fermion. Additional screening from the positive fermions together with use of an accurate local field factor naturally introduces charge density waves in addition to the $q=0$ instability that occurs when the bulk modulus equals zero. The electron-positive fermion gas is completely specified by the density $r_s$ and the mass ratio $M/m$. Although the problem and model can be simply stated, the resulting phase diagram is complex and not fully understood. The results of the simple model are exact formulas, and are in close agreement with earlier numerical results obtained using density functional theory in their region of overlap. Using these results, the electron-electron, positive fermion-positive fermion and electron-positive fermion many body effective interactions are calculated in the following paper. For conditions close to the charge density wave, the positive fermion contribution to the electron-electron interaction, which is attractive and the source of the superconductivity, becomes large and significantly enhances the superconducting transition temperature, as well as leading to a large $T^2$ contribution to the normal state electrical resistivity.