Lifshitz transitions and crystallization of fully-polarised dipolar Fermions in an anisotropic 2D lattice

We consider a two dimensional model of non-interacting chains of spinless fermions weakly coupled via a small inter-chain hopping and a repulsive inter-chain interaction. The phase diagram of this model has a surprising feature: an abrupt change in the Fermi surface as the interaction is increased. We study in detail this so called meta-nematic transition, and show that the well-known $2 1/2-$order Lifshitz transition is the criticalndpoint of this first order quantum phase transition. Furthermore, in the vicinity of the endpoint, the order parameter has a non-perturbative BCS form. We also study a competing crystallization transition in this model, and derive the full phase diagram. This physics can be demonstrated experimentally in dipolar ultra-cold atomic or molecular gases. In the presence of a harmonic trap, it manifests itself as a sharp jump in the density profile.