The semi-classical limit of large fermionic systems

We study a system of $N$ fermions in the regime where the intensity of the interaction scales as $1\mathrm{/}N$ and with an effective semi-classical parameter $\mathrm{\hslash }=N^{-1\mathrm{/}d}$ where $d$ is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas-Fermi minimizers in the limit $N\to \mathrm{\infty }$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti-Hewitt-Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.