Ground State Energy of the Low Density Fermi Gas

TL;DR

This paper rigorously derives the asymptotic formula for the ground state energy of a dilute low-density Fermi gas, confirming longstanding physics predictions using advanced mathematical techniques.

Contribution

It provides a rigorous mathematical derivation of the ground state energy formula for dilute Fermi gases, extending previous heuristic and perturbative results.

Findings

Derived the 3D ground state energy asymptotics for fermions

Extended the derivation to 2D systems with logarithmic corrections

Confirmed the validity of the known energy formulas for dilute gases

Abstract

Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of $N$ fermions interacting with a short-range, positive potential of scattering length $a$. For spin 1/2 fermions, this is $E \sim E^0 + (\hbar^2/2m) 2 \pi N \rho a$, where $E^0$ is the energy of the non-interacting system and $\rho$ is the density. A similar formula holds in 2D, with $\rho a$ replaced by $\rho /|\ln(\rho a^2)|$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.