Gross–Neveu model

The Gross–Neveu (GN) model is a quantum field theory model of Dirac fermions interacting via four-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu as a toy model for quantum chromodynamics (QCD), the theory of strong interactions. It shares several features of the QCD: GN theory is asymptotically free thus at strong coupling the strength of the interaction gets weaker and the corresponding ${\displaystyle \beta }$ function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with ${\displaystyle \mathbb {Z} _{2}}$ chiral symmetry breaking, and in the large number of flavor (${\displaystyle N\to \infty }$) limit, GN theory behaves as t'Hooft's large ${\displaystyle N_{c}}$ limit in QCD.

It consists of N Dirac fermions ${\displaystyle \psi _{1},\psi _{2},\cdots ,\psi _{N}}$. The Lagrangian density is

${\displaystyle {\mathcal {L}}={\bar {\psi }}_{a}\left(i\partial \!\!\!/-m\right)\psi ^{a}+{\frac {g^{2}}{2N}}\left[{\bar {\psi }}_{a}\psi ^{a}\right]^{2}}$.

Einstein summation notation is used, ${\displaystyle \psi ^{a}}$ is a two component spinor object and ${\displaystyle g}$ is the coupling constant. If the mass ${\displaystyle m}$ is nonzero, the model is massive classically, otherwise it enjoys a chiral symmetry.

This model has a U(N) global internal symmetry. If one takes N=1 (which permits only one quartic interaction) and makes no attempt to analytically continue the dimension, the model reduces to the massive Thirring model (which is completely integrable).

It is a 2-dimensional version of the 4-dimensional Nambu–Jona-Lasinio model (NJL), which was introduced 14 years earlier as a model of dynamical chiral symmetry breaking (but no quark confinement) modeled upon the BCS theory of superconductivity. The 2-dimensional version has the advantage that the 4-fermi interaction is renormalizable, which it is not in any higher number of dimensions.