Long Range Order in a Euclidean Gross-Neveu model on the lattice

Abstract: The Gross–Neveu (GN) model is a quantum field theory in $1+1$ dimensions describing $N$ massless Dirac fermions interacting through an attractive four-fermion coupling. Introduced by Gross and Neveu \cite{PhysRevD.10.3235} as a toy model for QCD, it shares two of its key features: asymptotic freedom and dynamical mass generation via spontaneous breaking of a $\mathbb{Z}_2$ chiral symmetry, allowing the fermion bilinear $(\overline{\psi}\psi)(x)$ to acquire a non-zero expectation value.

In this talk, we rigorously prove that a Euclidean lattice formulation of the Gross–Neveu model introduced by Cohen, Elitzur and Rabinovici exhibits long-range order in the $\mathbb{Z}_2$-charged fermion bilinear $\overline{\psi}\psi$ for sufficiently large $N$ in two spacetime dimensions.

The proof relies on reflection positivity of the bosonized measure obtained via a Hubbard–Stratonovich transformation of the fermionic action and, in particular, on chessboard estimates in the spirit of Fröhlich and Lieb (1978).

Joint work with Simone Fabbri (SISSA)

MathematicsPhysics

Audience: researchers in the topic

Probability, Statistical Mechanics and Quantum Fields

Series comments: Paweł Duch (EPFL) will give a mini-course on "Singular Stochastic PDEs", in the period March 9-20, 2026.