I'm reading papers with numerical lattice results that have this quantity called 'average plaquette' but I can't find a good definition anywhere. I know what a plaquette is, the smallest loop in a lattice. But that's not a number. There's no average plaquette to be found in my books on lattice either.

My guess is this is average of energy or action of all plaquette. But I could use an actual definition.

  • $\begingroup$ Sounds like a way to get through physics. $\endgroup$ – jjack Dec 24 '17 at 20:20

The smallest loop that you mention is an SU($N$) matrix (with $N=3$ for QCD). The real part of the trace of that matrix is the number in question. The average is taken over all lattice sites and orientations, so averaging over $6V$ numbers for lattice volume $V$ (in $d=4$ dimensions, where $6=d(d-1)/2$)).

It's common practice to refer to the trace itself as "the plaquette" rather than "the trace of the plaquette". You can see an example of this dynamic by comparing Eqs. 6.7 and 7.1 in Gupta's lecture notes hep-lat/9807028: Eq. 6.7 is the number while Eq. 7.1 is the matrix. (Note that you need the trace for gauge invariance.)

This really should be in lattice textbooks, but it may be considered so trivial that it isn't spelled out in detail. Another place you could look for precise definitions would be lattice codes that compute the plaquette, such as this routine in the MILC code.

  • $\begingroup$ Thanks! It's not the first time I miss something for being considered too obvious to be stated. I'm a very literal kind of person, maybe I would be more suited studying math instead :P $\endgroup$ – Diego Dec 25 '17 at 15:02

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