Berezinskii-Kosterlitz-Thouless phase transitions with long-range couplings

The Berezinskii-Kostelitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry-breaking, where a quasi-ordered phase, characterized by the power law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature $T_{\rm BKT}$. In this paper, we consider the effect of long-range decaying couplings $\sim r^{-2-\sigma}$ on this phenomenon. After pointing out the relevance of this non-trivial problem, we discuss the phase diagram, which is far richer with respect to the short-range interactions case. It features -- for $7/4 T_{\rm BKT}$. The transition temperature $T_c$ displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in $2D$ atomic, molecular and optical quantum systems.