# What are clover fermions?

I've seen the term been used quite a lot when reading about lattice gauge theory calculations. So far what I've gathered is the following, from this source [1].

Lorentz invariance of the action is broken on a discretized lattice. When calculating any quantity on the lattice, the continuum limit $$a\rightarrow 0$$ must be extracted, where $$a$$ is the lattice spacing. However this convergence is often slow, with $$O(a)$$ error terms, so one way to speed it up is to introduce an $$O(a)$$ irrelevant operator to the Lagrangian density, and tune its coefficient so that the lattice artifacts disappear in certain physical quantities (masses, cross sections, etc.).

$$\mathcal{L}\rightarrow \mathcal{L}+\int c_{\textrm{sw}}\mathcal{L}_c\,$$

$$\mathcal{L}_c=-\frac{a}{4} \bar \psi \sigma_{\mu\nu}\psi F^{\mu\nu}$$

The tuneable coefficient $$c_{\textrm{sw}}$$ is called the Sheikholeslami-Wohlert coefficient, and is a function of the lattice spacing through the gauge coupling $$g$$. The correction term $$\mathcal{L}_c$$ is called the "clover term".

So this would tell me what the clover-term is, the clover-improved action, but what is a clover fermion?

[1] "Clover fermions in the adjoint representation and simulations of supersymmetric Yang-Mills theory" arXiv:1311.6312 [hep-lat]

A clover fermion is a fermion described by the Wilson fermion action plus the clover-term. Clover fermion is just a short-hand version of Wilson clover fermion. Other terms which describe the same fermion discretization are clover-improved Wilson fermion or $$\mathrm{O}(a)$$ improved Wilson fermion.

• What is the Wilson fermion action? Is it different from the standard renormalizable fermion action $\int i\bar\psi \gamma^\mu D_\mu \psi$? Commented Aug 5, 2021 at 16:15
• The Wilson Dirac operator is defined in equation (2.7) of the reference you linked (the corresponding action in equation (2.6)). The original reference is Phys.Rev.D 10 (1974) 2445-2459.
–  Mio
Commented Aug 6, 2021 at 10:25
• Thanks. I have a question. In the paper I referenced, what is the continuum analog of the Dirac-Wilson operator $D_W(y,x)$? I would guess that it should be the standard bilocal version of the standard Dirac operator $\delta(y,x)\gamma^\mu D_\mu (x)$ where $D_\mu (x)$ is the covariant derivative with respect to $x$ Commented Aug 6, 2021 at 15:48
• That is correct. The Wilson Dirac operator approaches the continuum limit with cutoff effects of $\mathrm{O}(a)$ and without any doublers (which is the initial motivation for the Wilson term).
–  Mio
Commented Aug 7, 2021 at 17:08