Gross–Neveu model
Quantum theory in 1+1 dimensions / From Wikipedia, the free encyclopedia
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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[1] 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 function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with chiral symmetry breaking, and in the large number of flavor () limit, GN theory behaves as t'Hooft's large limit in QCD.[2]
It consists of N Dirac fermions . The Lagrangian density is
- .
Einstein summation notation is used, is a two component spinor object and is the coupling constant. If the mass 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).[3]
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.