Renormalization Group for a Mott Quartic fixed point

Fermi liquid theory can be viewed as a fixed point because of the stability of the Fermi surface to short-range repulsive interactions. The only instability is pairing in the Cooper channel. In the context of the cuprates, a similar question arises: Is there a fixed point that captures Mott physics and the underlying superconducting instability? This work builds on recent scaling and symmetry arguments which establish that Mott physics is controlled by a fixed point that breaks the $Z_2$ symmetry of a Fermi liquid[Huang extit{et al}., Nat. Phys. extbf{18}, 511 (2022)]. We demonstrate this here by a renormalization argument starting from the Hatsugai-Kohmoto model as the zeroth-order action. Considering a d-dimensional(d>2) system with a Mott transition controlled by the HK interaction, we show that at tree-level scaling analysis, the only relevant perturbation is of the superconducting(BCS) type. We also derive the RG equations at one-loop for the two-particle vertex function and show that indeed the HK model represents a true low-temperature fixed point for Mott physics. That is, interactions of the Hubbard type (modulo magnetism) do not destroy the Luttinger surface.