I really need to get into a more regular habit, posting things here. Of course, I've been busy, but it'd be nice to get into a regular habit.

I've been meaning to continue discussing various ways of simulating fermions in lattice QCD, so today I'll say a something about staggered fermions. Way back when, I talked about the basic problem with fermions. If you take the Dirac equation, and construct a "naive" approximation to it, you find that the theory you though was describing a single quark is actually describing 16 identical copies of quark. Each of these copies is known as a "taste". So we want to simulate one taste, but, at least naively, we get 16, that's the basic problem.

Now one way to get around this problem is to use something called Wilson fermions. I covered this in another post, so I won't say much about it, the basic idea is you construct a "non-naive" approximation in a clever way. In this setup you still have

16 tastes, but the are no longer all identical. Fifteen of them get masses that are

inversely proportional to the lattice spacing. The disadvantage here is you maul something called "chiral symmetry" which bites you when you try to simulate with light quarks.

There is another approach you can use to reduce the number of tastes, called "staggering". It's called that because you can think of it as putting the four spin components of the Dirac spinor on different sites of you lattice. There is a somewhat easier way to understand this. The Dirac action looks something like this

$$

\sum_x \bar{\psi} (Dslash + m) \psi

$$

Now \psi is a four component spinor and (Dslash + m) is some matrix that acts on it.

What would happen if we diagonalized this matrix?

If you know some quantum field theory you can do this exercise yourself, if not, take my word for it, when you diagonalise the matrix you get

$$

\sum_x \sum__{i=1,4} \chi_i^{*} (f(x) D + m) \chi_i

$$

Where \chi is now a *one* component field and f(x) is some function that depends on x. But notice, for each "i" in the sum, nothing changes. That is, we've produced four copies of the same thing. So what we'll do is just keep one. This is "staggered fermions". This procedure reduces the number of tastes from 16 to 4, which is better. There's another trick, to reduce the number of tastes to 1, but I'll save that for another day.

Now why might we like staggered fermions? Well, it turns out, they're much faster than Wilson fermions. Way faster. What's the reason? The first reason is that we're working with one component fields, rather than 4 component fields, so we save a factor of four right there. But the real savings comes when thinking about the chiral symmetry. Chiral symmetry is a symmetry between left and right handed fermions. For massless fermions it's exact, for small masses, it's violated in a small way.

No matter what the fermion mass is though chiral symmetry has the effect of "protecting" the fermion mass for additive shifts. In quantum field theory your particle properties get changed by the interactions. In a theory with chiral symmetry the mass can only be changed by multiplication by a constant. So if you start with some mass m0, the interactions can change it to Z*m0, but that's it. And Z is typically a number around 1. Now in a theory without chiral symmetry (like Wilson fermions) you can have an additive mass shift, so m0 might get changed to

Z*m0 + M1

where M1 can be large, and have either sign.

Now what does this have to do with the speed of lattice QCD simulations? Well, it is very important, because the speed at which the fermions can be simulated is inversely proportional to the interaction mass. What can happen with Wilson fermions is that the interaction mass gets very small (if M1 gets close to -Z*m0). Then the amount of time you spend processing shoots up. This never happens with staggered quarks. What's worse, the additive bit M1, can actually make the interaction mass go negative! This is even worse, because negative mass fermions have properties you don't want. Again, this cannot happen with staggered quarks.

It's not all sunshine though. Staggered quarks have a deep dark secret, which I will return to in another post.

## 2 comments:

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