Four - Fermi Theories in Fewer Than Four Dimensions

Four-fermi models in dimensionality $2<d<4$ exhibit an ultra-violet stable renormalization group fixed point at a strong value of the coupling constant where chiral symmetry is spontaneously broken. The resulting field theory describes relativistic fermions interacting non-trivially via exchange of scalar bound states. We calculate the $O(1/N_f)$ corrections to this picture, where $N_f$ is the number of fermion species, for a variety of models and confirm their renormalizability to this order. A connection between renormalizability and the hyperscaling relations between the theory's critical exponents is made explicit. We present results of extensive numerical simulations of the simplest model for $d=3$, performed using the hybrid Monte Carlo algorithm on lattice sizes ranging from $8^3$ to $24^3$. For $N_f=12$ species of massless fermions we confirm the existence of a second order phase transition where chiral symmetry is spontaneously broken. Using both direct measurement and finite size scaling arguments we estimate the critical exponents $\beta$, $\gamma$, $\nu$ and $\delta$. We also investigate symmetry restoration at non-zero temperature, and the scalar two-point correlation function in the vicinity of the bulk transition. All our results are in excellent agreement with analytic predictions, and support the contention that the $1/N_f$ expansion is accurate for this class of models.