# Interacting fermions on a lattice

My rough understanding about lattice simulations of bosonic quantum field theories is that the partition function can be approximated by explicitly summing over a large number of field configurations, chosen by some Monte Carlo algorithm for instance. But is there a similar way to simulate fermions in order to get at nonperturbative effects?

If an interaction is quadratic in the fermionic fields, like the Yukawa interaction $\phi \bar{\psi} \psi$, we can evaluate the fermionic path integral analytically for each particular field configuration $\phi$ in our Monte Carlo algorithm. But what if there is something not quadratic like a four-fermion interaction $(\bar{\psi}\psi)^2$?

I know that if the number of lattice sites is finite the perturbation series must end at some finite order (since there are a finite number of Grassmann variables), so in principle this can be solved `nonperturbatively' by simply summing over all terms. But of course that would be extremely inefficient, so how is this treated in practice?

The standard trick is partial bosonization, a.k.a. the "Hubbard-Stratonovich" trick. Consider ${\cal L}=g(\bar{\psi} \psi)^2$. Introduce a dummy field $\sigma$ with purely Gaussian lagrangian ${\cal L}_\sigma=-\frac{1}{g}\sigma^2$. You can always insert a factor 1 in the path integral $$1=\frac{1}{Z}\int D\sigma \exp(iS_\sigma).$$ Now shift the scalar field $\sigma\to\sigma-g\bar\psi\psi$. Then the quartic interaction among the fermions disappears and you are left with a Yukawa interaction ${\cal L}=2\sigma\bar{\psi}\psi$. There are many different versions and implementations of this general idea which you will find discussed at length in the quantum Monte Carlo literature. The problem is that for the "wrong sign" four-fermion interaction (repulsve) you will end up with imaginary Yukawa couplings -- the infamous sign (complex phase) problem.