Self-consistent harmonic approximation with non-local couplings

Abstract

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 $\sigma$, 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 for the lower critical threshold ${\sigma }^{\ast }=2$, above which the traditional BKT transition persists in spite of the LR couplings.

Submitted for the special issue "Quantum Long-Range Interactions" in Europhysics Letters, 4 figures

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