linear dimension $L$). The lattice Green’s function $G(r_{ij})$ solves the equation

$$(\Delta^2 - { 1 \over {\lambda^2}}) G(r_{ij})=-2\pi\delta_{r_{ij},0}$$

where $\Delta^2$ is the discrete lattice Laplacian and $\lambda$ is the screening length which, in normal case of no screening, is set to an infinity. For the case of usual Villain transformation of UFXY model, we have $\lambda = \infty$. But it is included in this equation for generality. Since, in this work, we restrict our attention to only square lattice with periodic boundary conditions, $G(r)$ is given by

$$G(\vec{r})={\pi \over N} \sum_{\vec k \neq 0} { e^{i\bf{k} \cdot r}-1 \over 
{2 - \cos k_x -\cos k_y + 1/\lambda^2} } ,$$

where $\bf{k}$ are the allowed wave vectors with $k_{\mu} = (2 \pi n_{\mu}/ L)$, with $n_{\mu} = 0, 1, \dots, L-1 $. In the case of infinite screening length, for large separation $r$, one gets $G(\vec r)\simeq -\ln r$ [@jrl]. In this work, we consider the limiting case of $ \lambda \rightarrow \infty $ only.

In our MC simulations, the initial disordered random configuration is updated according to the standard Metropolis algorithm by selecting a positive charge at random and moving it over to one of the [*nearest neighbor (NN)*]{} or 