There're a bunch of models of fermions with quartic self-interactions. There's an introduction from this wikipedia page.
For example, one can construct the Soler model of self-interacting Dirac fermions:
$$\mathcal{L}=\bar{\psi}(i\displaystyle{\not}\partial-m)\psi+\frac{g}{2}(\bar{\psi}\psi)^{2}.$$
But these fermionic fields in the classical Lagrangian should be Grassmann number valued, i.e. $\left\{\bar{\psi},\psi\right\}=0$, and $\psi\psi=\bar{\psi}\bar{\psi}=0$. Then the quartic interaction automatically vanish. So in the partition function
$$\mathcal{Z}=\int\mathcal{D}\psi\int\mathcal{D}\bar{\psi}\,e^{-S\,[\bar{\psi},\psi]}$$
it seems that the quartic term makes no contribution.
Am I misunderstanding anything?
