The Gross–Neveu model (GN) is a quantum field theory model of Dirac fermions interacting via a four-fermion interactions. It was introduced in 1974 by David Gross and André Neveu as a toy model for quantum chromodynamics (QCD), the theory describing the strong interaction. In 1 spatial and 1 time dimension, the Gross–Neveu shares several properties with QCD: First, it is asymptotically free, as the effective interaction strength decreases with increasing energy. Second, the theory has a dynamical mass generation mechanism with Z 2 {\displaystyle \mathbb {Z} _{2}} chiral symmetry breaking. It is made using a finite, but possibly large number, N , {\displaystyle \ N\ ,} of Dirac fermion wave functions ψ 1 , ψ 2 , … , ψ N {\displaystyle \psi _{1},\psi _{2},\ldots ,\psi _{N}} , indexed below by Latin letter a . {\displaystyle \ a~.}
The model's Lagrangian density is
L = ψ ¯ a ( i ∂ / − m ) ψ a + g 2 2 N [ ψ ¯ a ψ a ] 2 , {\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 ψ a {\displaystyle \ \psi ^{a}\ } is a two component (left / right) spinor and g {\displaystyle \ g\ } is the interaction's coupling constant. If the mass m {\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 N = 1 {\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.
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