Vortex screening and the fate of the BKT transition with long-range couplings

The stability of the Berezinskii–Kosterlitz–Thouless phase against long-range interactions remains an open problem. We address this question in the two-dimensional XY model with couplings decaying as $r^{-2-\sigma }$, by developing a real-space renormalization-group treatment near the Gaussian manifold of quasi-long-range ordered (QLRO) states. We retain the full many-body vortex interaction and show that, at leading order in the fugacity, screening by small vortex–antivortex pairs preserves its Gaussian-average representation without an expansion in the long-range coupling. Including spin waves perturbatively then yields a local stability criterion: a stable QLRO interval exists only for $\sigma >7\mathrm{/}4$. We also investigate a Wilson–Fisher-like fixed point at finite coupling for $\sigma <7\mathrm{/}4$, for which the retained flow predicts ${\nu }^{-1}\sim \sqrt{7\mathrm{/}4-\sigma }$ as it merges with the Gaussian line.