Self-consistent harmonic approximation with non-local couplings

TL;DR

This paper develops a self-consistent harmonic approximation for the 2D XY model with non-local interactions, analyzing the persistence of the BKT transition under power-law couplings decaying as 1/r^{2+σ}.

Contribution

It derives a general variational equation for non-local couplings and establishes that the BKT transition persists for all σ>2, providing an upper bound for the critical threshold.

Findings

BKT transition occurs for any σ>2.

Provides an upper bound σ* = 2 for the transition persistence.

Formulates an ansatz for variational couplings in non-local interactions.

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.