Villain model with long-range couplings

The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to understand the physics of topological defects two-dimensional $XY$ model, as two models share same symmetries and are in universality class. long-range counterpart has been recently shown exhibit non-trivial critical behavior, with complex phase diagram including range values power-law exponent couplings decay, $\sigma$, which there magnetized, disordered (arXiv:2104.13217). Here we address issue whether behavior can be described by Villain model. After introducing suitable generalization couplings, derive set renormalization-group equations for vortex-vortex potential, differs from one signaling that decoupling spin-waves no longer justified this regime. main results $\sigma<2$ Remarkably, within large region its diagram, found behave similarly one-dimensional Ising $1/r^2$ interactions.