Gross–Neveu model

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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 as a toy model for quantum chromodynamics, the theory of strong interactions.

It consists of N Dirac fermions, ψ₁, ..., ψ_N. 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. When N=1 it reduces to the integrable Thirring model.

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 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.

The Gross–Neveu model is named after David Gross and André Neveu.^[1] They 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 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.

References

1. ^ Gross, David J. and Neveu, André (1974). "Dynamical symmetry breaking in asymptotically free field theories". Phys. Rev. D 10: 3235--3253. doi:10.1103/PhysRevD.10.3235. 

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