Self-consistent harmonic approximation in presence of non-local couplings(a)

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