Four-Fermi Theories in Fewer Than Four Dimensions

Abstract

Four-fermi models in dimensionality 2 < d < 4 exhibit an ultraviolet 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 ciritical 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 β, γ, ν, and δ. 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 model.