Self-consistent harmonic approximation in presence of non-local couplings

We derive the self-consistent harmonic approximation for the 2D XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance r as $\propto 1/r^{2+\sigma}$ in order to investigate the robustness, at finite σ, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit $\sigma \to \infty$ . We propose an ansatz for the functional form of the variational couplings and show that for any $\sigma>2$ ?> the BKT mechanism occurs. The present investigation provides an upper bound $\sigma^\ast=2$ for the critical threshold $\sigma^\ast$ above which the traditional BKT transition persists in spite of the non-local nature of the couplings.