evanberkowitz
commented
12 months ago This was closed by #53.
However, the implementation is a bit weird in the sense that the correlator isn't in the A1 representation in expectation.
The reason, I think, is that I implemented the action density as suggested above: assign to each site just the action from the links 'coming out of' the site and ignoring the links 'landing on' the site.
A more symmetric thing to do would be for a site to claim half of every link touching it. That can be left for the future, if ever.
In the Villain formulation, we can write the action as $
S = \frac{\kappa}{2} \sum_\ell (d\phi - 2\pi n)_\ell^2$ which we can write as $S = \frac{\kappa}{2} \sum_x \sum_{\mu \in \{\hat{t},\hat{x}\}} (d\phi - 2\pi n)_{x,\mu}^2$. This makes it really look like $(\partial \phi)^2$ summed over spacetime.Now we can move κ inside the spatial sum and promote it to a local variable $
S = \frac{1}{2} \sum_x \kappa_x \sum_{\mu \in \{\hat{t},\hat{x}\}} (d\phi - 2\pi n)_{x,\mu}^2$. We will functionally differentiate with respect to κ at two (potentially) different sites to compute action-density correlations, and then plug $\kappa_x = \kappa$ so that we can sample according to the usual distribution.The same functional differentiation may be carried out on the Worldline side.
We should implement this observable!