Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent
z=1. This type of critical behavior can in principle be studied by deforming ordinary
z=1 conformal field theories (CFTs) by relevant vector operators breaking the rotational/Lorentz symmetry. In this short note, we consider a two-dimensional system of coupled minimal model CFTs
Mm,m+1 which realizes this perspective in a controlled fashion via Zamolodchikov's large
m expansion. The model turns out to exhibit interesting properties, including a manifold of interacting Lifshitz fixed points and emergent rotational symmetry in the infrared.
Anisotropic dipole-dipole interactions between ultracold dipolar fermions break the symmetry of the Fermi surface and thereby deform it. Here we demonstrate that such a Fermi surface deformation induces a topological phase transition---the so-called Lifshitz transition---in the regime accessible to present-day experiments. We describe the impact of the Lifshitz transition on observable quantities such as the Fermi surface topology, the density-density correlation function, and the excitation spectrum of the system. The Lifshitz transition in ultracold atoms can be controlled by tuning the dipole orientation and, in contrast to the transition studied in crystalline solids, is completely interaction driven.