  We study in this paper the sine-Gordon model using functional Renormalization
Group (fRG) at Local Potential Approximation (LPA) using different RG schemes.
In $d=2$, using Wegner-Houghton RG we demonstrate that the location of the
phase boundary is entirely driven by the relative position to the Coleman fixed
point even for strongly coupled bare theories. We show the existence of a set
of IR fixed points in the broken phase that are reached independently of the
bare coupling. The bad convergence of the Fourier series in the broken phase is
discussed and we demonstrate that these fixed-points can be found only using a
global resolution of the effective potential. We then introduce the methodology
for the use of Average action method where the regulator breaks periodicity and
show that it provides the same conclusions for various regulators. The behavior
of the model is then discussed in $d\ne 2$ and the absence of the previous
fixed points is interpreted.
