2.1 Metropolis update

In the sumulation we need to store only the phase of the field $\theta$ as a double, and we update the configuration extracting a number $-1<\delta(x)<1$ for each point $x$ and considering the new configuration $$\phi'(x)=e^{i(\theta(x)+\delta(x))}\,,$$ then we compute the variation of the action

$$\Delta S=S[\phi']-S[\phi]$$