Exactly Solvable Model for Isospin $S=3/2$ Fermionic Atoms on an Optical Lattice

We propose an exact solution of a model describing a low energy behavior of cold isospin $S=3/2$ fermionic atoms on a one-dimensional optical lattice. Depending on the band filling the effective field theory has a form of a deformed Gross-Neveu model with either $\mathrm{O}(7)\times {\mathbb{Z}}_2$ (half filling) or $\mathrm{U}(1)\times \mathrm{O}(5)\times {\mathbb{Z}}_2$ symmetry.