====== Gross-Neveu model ======

{{tag>3d}}

The Gross-Neveu model is a theory of $N$ interacting fermions defined by the Lagrangian density
\[
\mathcal L = \bar\psi i \gamma^\mu \partial_\mu \psi + \frac{g^2}{2N} (\bar\psi \psi)^2
\text.
\]

This model is usually studied in two dimensions, where it is asymptotically free for any $N$. However, for $2 < d < 4$, the theory is asymptotically safe: [[https://arxiv.org/pdf/1011.1456.pdf]]. Thus, there's a nontrivial CFT corresponding to the UV fixed point. This page is about that CFT.

===== Large-N Limit =====

[[https://www.sciencedirect.com/science/article/pii/037026939291265B?via%3Dihub]]

===== Other dimensions =====

It is more common to study the Gross-Neveu model in two dimensions, where it is asymptotically free (so that the fixed point coincides with the theory of $N$ free fermion fields).

In four dimensions this model is trivial.

===== See also =====

[[gross_neveu_yukawa_model]]

===== External links =====

  * [[https://en.wikipedia.org/wiki/Gross%E2%80%93Neveu_model|Wikipedia]] only talks about the 2d model.