Gross–Neveu model

The Gross–Neveu model (GN) 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 is made using a finite, but possibly large number, ${\displaystyle \ N\ ,}$ of Dirac fermion wave functions ${\displaystyle \psi _{1},\psi _{2},\ldots ,\psi _{N}}$, indexed below by Latin letter ${\displaystyle \ a~.}$

The model's Lagrangian density is

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

where the formula uses Einstein summation notation. Each wave function ${\displaystyle \ \psi ^{a}\ }$ is a two component (left / right) spinor and ${\displaystyle \ g\ }$ is the interaction's coupling constant. If the mass ${\displaystyle \ m\ }$ is zero, the model is chiral symmetric type, otherwise, for non-zero mass, it is classical mass type.

This model has a U(N) global internal symmetry. If one takes ${\displaystyle \ N=1\ }$ (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.