Investigating the Homogeneous Electron Gas with an Approximation in Lattice Field Theories.

The Homogeneous Electron Gas (HEG) is a model in which electrons freely move around in the presence of a background positive charge density. This model is often used by condensed matter physicists as a simpler model of their more complex system, so it has been studied extensively. Physicists discovered that at high electron densities, this model would be a fluid, but at low electron densities, it resembled a crystalline structure. However, the region in-between has been much harder to probe - both theoretically and experimentally. One idea was a convincing argument in 1+1 dimensions that there must exist a colloidal or gel-like phase at intermediate densities, which meant that the fluid turned into a crystal through a second order phase transition. This phenomenon is invisible when using perturbation theory, so an alternative approach was proposed: an approximation to the Schwinger-Dyson equations for large fermionic spin components N. In this project, we test how accurate this approximation is on 2 x 2 and 3 x 3 lattices. We derive the relevant equations, consisting of 1 and 2 point correlation functions, which we numerically calculate using the Markov Chain Monte Carlo numerical method. Plugging these into the equations, we find a match of order 10-3, which strongly suggests that the approximation is successful. We are now studying the actual HEG model from both the fluid and crystalline phases to check if a second order phase transition appears from the corresponding equations.