Gross–Neveu model
The Gross–Neveu 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, the theory of strong interactions.
It consists of N Dirac fermions, ψ1, ..., ψN. The Lagrangian density is
using the Einstein summation notation where g is the coupling constant. If the mass m is nonzero, the model is massive classically, otherwise it enjoys a chiral symmetry.
This model has an U(N) global internal symmetry. Note that it does not reduce 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.
Gross and Neveu studied this model in the large N limit, expanding the relevant parameters in a 1/N expansion. After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate acquires a vacuum expectation value (VEV) and as a result the fundamental fermions become massive. They find that the mass is not analytic in the coupling constant g. The vacuum expectation value spontaneously breaks the chiral symmetry of the theory.
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