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| Gross–Neveu model | |
|---|---|
| Name | Gross–Neveu model |
| Field | Quantum field theory |
| Introduced | 1974 |
| Authors | David Gross; André Neveu |
| Notable results | Asymptotic freedom; dynamical mass generation; large-N solvability |
Gross–Neveu model The Gross–Neveu model is a two-dimensional relativistic quantum field theory of interacting fermions introduced in 1974 by David Gross and André Neveu. It provided an exactly tractable laboratory for nonperturbative phenomena such as asymptotic freedom, dynamical symmetry breaking, and bound-state formation, and quickly influenced research in particle physics, condensed matter, and statistical mechanics. Major developments relating to the model involve work by Gerard 't Hooft, Alexander Polyakov, Kenneth Wilson, Sidney Coleman, and Michael Peskin, among others.
The model was proposed in the context of studies by David Gross and André Neveu seeking lower-dimensional analogues of Quantum Chromodynamics and testing ideas from Renormalization group analysis, inspired by earlier insights of Kenneth Wilson and Miguel Virasoro. It was motivated by attempts to understand asymptotic freedom discovered in four dimensions by Gross and Frank Wilczek and David Politzer, and to explore dynamical symmetry breaking in a simpler setting related to work by Yoichiro Nambu and Giovanni Jona-Lasinio. Early solutions and large-N techniques were influenced by methods introduced by Gerard 't Hooft and subsequent exact results were connected to studies by Sidney Coleman and Alexander Polyakov.
The original formulation describes N species of massless, two-component Dirac fermions in two-dimensional Minkowski space with a quartic, flavor-singlet interaction devised by David Gross and André Neveu. The Lagrangian density was written to mirror aspects of Quantum Chromodynamics and the Nambu–Jona-Lasinio model while remaining renormalizable in two dimensions, a strategy inspired by methods of Kenneth Wilson and Miguel Virasoro for simplified models.
The Gross–Neveu model possesses a global U(N) flavor symmetry (or O(N) in variants) analogous to flavor symmetries studied by Murray Gell-Mann and George Zweig, discrete chiral symmetries related to the work of Yoichiro Nambu, and Poincaré invariance as in quantum field theories analyzed by Richard Feynman and Julian Schwinger. The model exhibits conserved currents associated with flavor rotations and energy–momentum, with anomaly considerations paralleling studies by Stephen Adler and John Bell in other contexts.
The 1/N expansion for the model was developed using techniques pioneered by Gerard 't Hooft for large-N QCD and by Alexander Polyakov for two-dimensional field theories. At leading order in 1/N the model becomes solvable: saddle-point methods and auxiliary field redefinitions, similar to approaches used by Yoichiro Nambu and Giovanni Jona-Lasinio, yield gap equations and propagators analyzed by Coleman and later by Michael Peskin. Subleading 1/N corrections were computed using diagrammatic techniques employed in studies by Steven Weinberg and Lars Onsager in related contexts.
Renormalization of the model follows methods developed by Kenneth Wilson and John Kogut for lower-dimensional theories; the model is perturbatively renormalizable in two dimensions and exhibits asymptotic freedom as established in the original work of Gross and Neveu, paralleling asymptotic freedom in Quantum Chromodynamics discovered by Frank Wilczek and David Politzer. Renormalization group flows and beta functions were studied using techniques from Kenneth Wilson and the formalism employed in analyses by Politzer and Gerard 't Hooft.
The model demonstrates dynamical mass generation through spontaneous breaking of discrete chiral symmetry, an effect analogous to the mechanism proposed by Yoichiro Nambu for mass generation and related to the Nambu–Jona-Lasinio model studied by Giovanni Jona-Lasinio. Bound-state spectra, including meson-like excitations, were obtained using Bethe–Salpeter and S-matrix approaches similar to those applied by Salam, Weinberg, and Coleman; exact S-matrix constructions in two dimensions drew on work by Alexander Zamolodchikov and Luis A. Fernández in integrable models. Soliton and kink solutions relate to classical analyses by Mikhail Shifman and studies of two-dimensional solitons by Roman Jackiw.
The Gross–Neveu model has been used as a testbed in contexts ranging from lattice field theory studies by Kenneth Wilson and Michael Creutz to condensed-matter analogues in systems discussed by Philip Anderson and Nikolay Mott. It connects to integrable field theories studied by Alexander Zamolodchikov and Vladimir Fateev, to models of low-dimensional superconductivity influenced by John Bardeen and Leon Cooper, and to developments in holography and AdS/CFT inspired by Juan Maldacena and Edward Witten. Variants and relatives include the Thirring model analyzed by Walter Thirring, the Sine-Gordon model explored by Sidney Coleman, and the Nambu–Jona-Lasinio model linked to Yoichiro Nambu.
Category:Quantum field theory models