Exact Results in Quantum Theory & Gravity

We consider a simple model of interacting fermions in low dimensions, with the long-range attractive force giving rise to an all-to-all interaction (Kac limit). We discuss the stability conditions for the thermodynamics to exist, and find the grand canonical partition function exactly. It turns out that upon varying the embedding dimension, or, equivalently, the exponent in the dispersion relation, the equilibrium thermodynamics of the system varies dramatically and is accompanied by a first-order phase transition at low dimensions, which vanishes at the boundary dimension for thermodynamics to exist, where it is replaced by complete and exact bosonization. If time permits, we shall discuss extensions to the case of a spin-polarized dilute Fermi gas with an additional short-range repulsion. Joint work with Marek Napiórkowski.