{ "id": "ep6qxe73pf", "slug": "gross-neveu-model", "title": "Gross–Neveu model", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.quantum", "label": "Quantum physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.quantum" }, { "id": "physical.physics.quantum.field-theory", "label": "Quantum field theory", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.quantum.field-theory" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.physics", "label": "Non-geographic · 1946 to 2000: Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" } ] } ], "excerpt": "The Gross–Neveu model is a quantum field theory of massless Dirac fermions interacting through a four-fermion contact term, introduced by David Gross and André Neveu in 1974.", "snippet": "The Gross–Neveu model is a quantum field theory of massless Dirac fermions interacting through a four-fermion contact term, introduced by David Gross and André Neveu in 1974.", "node": "physical.physics.quantum.field-theory", "markdown": "# Gross–Neveu model\n\nThe Gross–Neveu model is a quantum field theory of massless Dirac fermions interacting through a four-fermion contact term. It is asymptotically free, generates a fermion mass dynamically, and spontaneously breaks a discrete chiral symmetry, making it probably the simplest example of fermionic criticality.[1](https://journals.aps.org/prl/pdf/10.1103/trj9-r9j8) • [2](https://arxiv.org/pdf/2010.03441) • [3](https://link.aps.org/doi/10.1103/PhysRevD.109.096026) Because it shares asymptotic freedom with QCD while remaining solvable, it serves as a laboratory for quantum phase transitions in materials with gapless quasirelativistic electrons.[4](https://ar5iv.labs.arxiv.org/html/hep-th/0008175) • [3](https://link.aps.org/doi/10.1103/PhysRevD.109.096026)\n\n| Key fact | Statement |\n|---|---|\n| Content | N species of massless Dirac fermions in two spacetime dimensions with an attractive four-fermion (scalar-scalar) contact interaction[1](https://journals.aps.org/prl/pdf/10.1103/trj9-r9j8) • [2](https://arxiv.org/pdf/2010.03441) |\n| Symmetry | For vanishing bare mass, a discrete Z₂ chiral symmetry ψ → γ₅ψ, spontaneously broken in the ground state[2](https://arxiv.org/pdf/2010.03441) |\n| Mass generation | To leading order in large N, dimensional transmutation gives \\( m = \\mu \\exp[-\\pi/g_{R}^{2}(\\mu)] \\) with \\( \\beta(g_{R}) = \\mu\\, dg_{R}/d\\mu = -g_{R}^{3}/(2\\pi) \\) in the stated convention[5](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/) |\n| Origin | Introduced by David J. Gross and André Neveu, Physical Review D 10, 3235 (1974)[6](https://doi.org/10.1103/physrevd.10.3235) |\n| Solvability | Exactly solvable at large N; integrable, with spectrum and complete S-matrix known for any N[1](https://journals.aps.org/prl/pdf/10.1103/trj9-r9j8) |\n| Applications | Variants describe quantum phase transitions in graphene, unconventional superconductors, and topological-insulator surfaces[3](https://link.aps.org/doi/10.1103/PhysRevD.109.096026) |\n\n## How it works\n\nThe model describes N species of Dirac fermions coupled by an attractive quartic scalar-scalar interaction of the form (ψ̄ψ)². When the bare mass m₀ vanishes, the Z₂ chiral symmetry ψ → γ₅ψ forbids any perturbative contribution to the condensate, so no mass term is allowed perturbatively.[2](https://arxiv.org/pdf/2010.03441) The four-fermion coupling is classically marginal in two dimensions, and the model is renormalizable; the coupling is asymptotically free, tending to zero in the ultraviolet while growing toward the infrared, and the ground state develops a dimensionful condensate even though the only dimensionful parameter is the ultraviolet regulator scale, which is dimensional transmutation.[2](https://arxiv.org/pdf/2010.03441) To leading order in large N, the resulting fermion mass is renormalization-group invariant and takes the exponential form \\( m = \\mu \\exp[-\\pi/g_{R}^{2}(\\mu)] \\), with beta function \\( \\beta(g_{R}) = \\mu\\, dg_{R}/d\\mu = -g_{R}^{3}/(2\\pi) \\) in the stated 1/N-normalized convention; at finite N the beta function and the mass-to-scale relation receive corrections, and the overall mass prefactor is regulator-dependent.[5](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/)\n\nA Hubbard–Stratonovich auxiliary scalar σ makes the large-N saddle transparent; in the discrete-chiral model its two saddle signs label symmetry-related vacua, and the saddle value equals the leading fermion pole mass.[5](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/) Two chiral choices exist: the original model with discrete symmetry ψ → γ₅ψ corresponds to λ = 0, m₀ = 0, while λ = 1, m₀ = 0 gives the Nambu–Jona-Lasinio-type model with continuous chiral symmetry, so in that case the Gross–Neveu model may be thought of as the NJL model in two dimensions.[4](https://ar5iv.labs.arxiv.org/html/hep-th/0008175) The original 1974 analysis showed that dynamical symmetry breaking occurs for any value of the coupling, producing a fermion mass, a scalar bound state, and, if the broken symmetry is continuous, a Goldstone boson.[6](https://doi.org/10.1103/physrevd.10.3235) In the continuous-chiral variant, however, long-wavelength fluctuations of the would-be Goldstone direction prevent a nonzero condensate at every finite N in infinite volume.[5](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/)\n\n## How it is done\n\nThe model is solved in the large-N limit by the Hartree approximation, replacing ψ̄ψ by its expectation value in the [Euler–Lagrange equation](https://www.edgepedia.org/euler-lagrange-equation); this is exact as N → ∞.[4](https://ar5iv.labs.arxiv.org/html/hep-th/0008175) With a discrete chiral symmetry the theory has two phases, a symmetric massless phase and a broken gapped phase.[7](https://www.sciencedirect.com/science/article/abs/pii/S0370157303002631) The \\( 1/N \\) expansion gives systematic corrections, but beyond leading order it develops logarithmic and pole singularities in \\( \\varepsilon \\) near two dimensions (\\( d = 2 - \\varepsilon \\)), so ultraviolet divergences must be handled at subleading order.[8](https://inspirehep.net/literature/14809) Finite-N corrections begin at subleading order, and the exponential mass mechanism is robust even though its regulator-level prefactor is not.[5](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/)\n\nThe theory is integrable: its spectrum, containing kinks and bound states of fermions, and its complete S-matrix are known for any N.[1](https://journals.aps.org/prl/pdf/10.1103/trj9-r9j8) In the large-N (mean-field) treatment, at finite temperature and chemical potential, three phase boundaries meet at a tricritical point: a chirally symmetric massless phase at high temperature, a homogeneous broken phase with dynamical mass M, and a low-temperature kink-antikink crystal phase; at finite N, a one-dimensional system at nonzero temperature cannot sustain true discrete-symmetry breaking, so these are not exact finite-N thermodynamic phases. Along the boundary to the crystal the kink amplitude jumps discontinuously while bulk thermodynamic observables remain continuous, because the perturbation is localized in space; a nonzero bare mass \\( m_{0} \\) breaks the chiral symmetry explicitly and turns the symmetric-phase boundary into a crossover.[9](https://journals.aps.org/prd/pdf/10.1103/PhysRevD.110.096012)\n\n## Origin\n\nThe model was introduced by [David J. Gross](https://www.edgepedia.org/david-j-gross) and [André Neveu](https://www.edgepedia.org/andre-neveu) in \"Dynamical symmetry breaking in asymptotically free field theories,\" [Physical Review D](https://www.edgepedia.org/physical-review-d) 10(10):3235–3253 (1974).[6](https://doi.org/10.1103/physrevd.10.3235) The paper analyzed two-dimensional massless fermion field theories with quartic interactions, which are asymptotically free and are expanded in powers of \\( 1/N \\), where \\( N \\) is the number of fermion field components. The problem addressed was dynamical symmetry breaking in asymptotically free theories, the same mechanism class as QCD, with which pure fermionic Gross–Neveu-type theories share asymptotic freedom.[4](https://ar5iv.labs.arxiv.org/html/hep-th/0008175)\n\n## Variants\n\nThree named Gross–Neveu–Yukawa (GNY) variants cover the main symmetry choices: the chiral [Ising model](https://www.edgepedia.org/ising-model) with a single real Z₂ order parameter, the chiral [XY model](https://www.edgepedia.org/xy-model) with continuous U(1) breaking described by a complex order parameter, and the chiral [Heisenberg model](https://www.edgepedia.org/heisenberg-model) with broken SU(2) symmetry.[10](https://bib-pubdb1.desy.de/record/396032/files/PRD96%282017%29096010.pdf?subformat=pdfa) The chiral Ising GNY model results from a Hubbard–Stratonovich decoupling of a four-Fermi interaction and lies in the same universality class as the purely fermionic Gross–Neveu model for \\( 2 < D < 4 \\); its Lagrangian is renormalizable in \\( D = 4 - \\varepsilon \\) dimensions, which is why GNY formulations are used for perturbative RG and lattice studies.[10](https://bib-pubdb1.desy.de/record/396032/files/PRD96%282017%29096010.pdf?subformat=pdfa) In two dimensions the scalar is auxiliary and the theory reduces to a quartic fermion interaction, whereas the four-dimensional GNY model requires an extra quartic scalar interaction for renormalizability.[11](https://arxiv.org/html/2507.22594v1) The Ising Gross–Neveu model has also become central to generalized Gross–Neveu universality classes with non-Abelian symmetry, where theories away from the phase transition are inequivalent.[12](https://www.imath.kiev.ua/~sigma/2021/064/sigma21-064.pdf) A lattice chiral Gross–Neveu model with both gx²(ψ̄ψ)² and gy²(ψ̄iγ₅ψ)² terms has a continuous chiral symmetry that carries a mixed 't Hooft anomaly with charge conservation, guaranteeing a massless mode.[13](https://link.springer.com/article/10.1007/JHEP06%282022%29019)\n\nOn the critical-behavior side, the O(N) GNY model has been renormalized to \\( O(\\varepsilon^{5}) \\) in \\( d = 4 - \\varepsilon \\), determining fermion and scalar anomalous dimensions, beta functions, and the scalar mass operator; for N = 2 the exponents connect to the semimetal-to-insulator transition in graphene, and improved estimates for \\( N = 1 \\) and \\( N = 5 \\) allow comparison with [conformal bootstrap](https://www.edgepedia.org/conformal-bootstrap) results.[11](https://arxiv.org/html/2507.22594v1)\n\n## Applications\n\nFor an eight-component spinor (\\( N = 2 \\)), the chiral Ising GNY model describes the semimetal–insulator quantum critical point of graphene, with a sublattice-symmetry-breaking charge-density-wave insulator as the ordered state.[10](https://bib-pubdb1.desy.de/record/396032/files/PRD96%282017%29096010.pdf?subformat=pdfa) The chiral XY variant for \\( N = 2 \\) is relevant to Kekulé valence-bond-solid transitions and superconducting states in graphene, while the chiral Heisenberg model describes a candidate interaction-driven transition to an antiferromagnetic spin-density-wave state, potentially accessible via biaxial strain.[10](https://bib-pubdb1.desy.de/record/396032/files/PRD96%282017%29096010.pdf?subformat=pdfa) At strong coupling the leading instabilities in these Dirac systems are toward Lorentz-invariant order parameters representing relativistic mass terms, which translate into broken-symmetry insulating or superconducting states of the original electrons.[3](https://link.aps.org/doi/10.1103/PhysRevD.109.096026) The \\( N = 2 \\) Gross–Neveu model also arises as the continuum description of polyacetylene.[2](https://arxiv.org/pdf/2010.03441)\n\nIn 2+1 dimensions, a lattice Gross–Neveu model shows a continuous transition from a gapless [Dirac semimetal](https://www.edgepedia.org/dirac-semimetal) to a gapped quantum anomalous Hall insulator at finite attractive coupling, with inversion and time-reversal symmetry spontaneously broken.[14](https://link.aps.org/doi/10.1103/45db-kr73) A sign-problem-free fermionic auxiliary-field quantum [Monte Carlo](https://www.edgepedia.org/monte-carlo) study of the repulsive regime found the O(4N) symmetry-breaking transition out of the Dirac semimetal to be weakly first order for \\( N = 2 \\), with the discontinuity and the critical coupling both growing with N.[14](https://link.aps.org/doi/10.1103/45db-kr73) Matrix product state simulations of the chiral lattice model show the continuous chiral symmetry reemerging at an infrared fixed point even at strong coupling, physics beyond the mean-field or leading-order \\( 1/N \\) treatment.[13](https://link.springer.com/article/10.1007/JHEP06%282022%29019)\n\n## Limitations and alternatives\n\nIn 3+1 dimensions, four-fermi interactions cannot be renormalizable, so the contact-interaction formulation is restricted to low dimensions; large-N Gross–Neveu-inspired constructions have been used to build solvable fermionic models in 3+1d instead.[15](https://export.arxiv.org/pdf/2302.08603v3.pdf) Even in 1+1 dimensions, the exponential mass formula's overall prefactor depends on the regulator, although the exponential mechanism itself does not.[5](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/) On the lattice, observables converge logarithmically slowly as the continuum limit is approached, making symmetry-preserving Hamiltonian constructions important.[2](https://arxiv.org/pdf/2010.03441) Finally, notable discrepancies persist between numerical and analytical studies, in particular between ε-expansion analyses and other approaches for certain GNY models such as Gross–Neveu–XY.[16](https://export.arxiv.org/pdf/2510.18875)\n\n## References\n\n1. [Condensates, Crystals, and Renormalons in the Gross-Neveu Model at Finite Density (Physical Review Letters)](https://journals.aps.org/prl/pdf/10.1103/trj9-r9j8)\n2. [Lattice regularisation and entanglement structure of the Gross-Neveu model (arXiv:2010.03441)](https://arxiv.org/pdf/2010.03441)\n3. [Spontaneous breaking of the symmetry in the Gross-Neveu model (Phys. Rev. D 109, 096026, 2024)](https://link.aps.org/doi/10.1103/PhysRevD.109.096026)\n4. [2D Model Field Theories at Finite Temperature and Density (Boris Ioffe Festschrift contribution)](https://ar5iv.labs.arxiv.org/html/hep-th/0008175)\n5. [The Gross–Neveu Model and Dynamical Mass Generation | QFT.org](https://qft.org/nonperturbative/strong-coupling-laboratories/gross-neveu-dynamical-mass/)\n6. [David J. Gross, André Neveu (1974). Dynamical symmetry breaking in asymptotically free field theories. Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields.](https://doi.org/10.1103/physrevd.10.3235)\n7. [Quantum field theory in the large N limit: a review (Physics Reports)](https://www.sciencedirect.com/science/article/abs/pii/S0370157303002631)\n8. [Ultraviolet Divergences in 1/N Expansions of Asymptotically Free Theories (INSPIRE record)](https://inspirehep.net/literature/14809)\n9. [Nonperturbative phase boundaries in the Gross–Neveu model from a stability analysis (Phys. Rev. D 110, 096012)](https://journals.aps.org/prd/pdf/10.1103/PhysRevD.110.096012)\n10. [Four-loop critical exponents for the Gross-Neveu-Yukawa models (PRD 96, 096010, 2017)](https://bib-pubdb1.desy.de/record/396032/files/PRD96%282017%29096010.pdf?subformat=pdfa)\n11. [Anomalous dimensions and critical exponents for the Gross-Neveu-Yukawa model at five loops (arXiv:2507.22594, 2025)](https://arxiv.org/html/2507.22594v1)\n12. [Generalized Gross–Neveu Universality Class with Non-Abelian Symmetry (SIGMA 17, 064, 2021)](https://www.imath.kiev.ua/~sigma/2021/064/sigma21-064.pdf)\n13. [JHEP06(2022)019 (link.springer.com)](https://link.springer.com/article/10.1007/JHEP06%282022%29019)\n14. [Phase transitions on the dark side of the Gross-Neveu model: Spontaneous symmetry breaking at repulsive coupling (Phys. Rev. B)](https://link.aps.org/doi/10.1103/45db-kr73)\n15. [A fully solvable model of fermionic interaction in 3+1d (arXiv:2302.08603)](https://export.arxiv.org/pdf/2302.08603v3.pdf)\n16. [Instabilities of a Generalized Gross-Neveu Quantum Criticality (arXiv:2510.18875, October 2025)](https://export.arxiv.org/pdf/2510.18875)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum field theory*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 21, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/gross-neveu-model", "markdown_url": "https://www.edgepedia.org/gross-neveu-model.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Gross–Neveu model\", Edgepedia (EdgeChat), https://www.edgepedia.org/gross-neveu-model. Edgepedia Community License 1.0.", "credit_md": "\"[Gross–Neveu model](https://www.edgepedia.org/gross-neveu-model)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/gross-neveu-model](https://www.edgepedia.org/gross-neveu-model). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Gross–Neveu model\", Edgepedia (EdgeChat), https://www.edgepedia.org/gross-neveu-model. Edgepedia Community License 1.0.", "speakable": "The Gross–Neveu model is a quantum field theory of massless Dirac fermions interacting through a four-fermion contact term, introduced by David Gross and André Neveu in 1974." }