The low-temperature phase in the two-dimensional long-range diluted XY model

Abstract

The critical behaviour of statistical models with long-range interactions exhibits distinct regimes as a function of $\rho$, the power of the interaction strength decay. For $\rho$ large enough, $\rho >{\rho }_{\mathrm{sr}}$, the critical behaviour is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\rho }_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2d XY model on the {\it diluted} graph, a sparse graph obtained from the 2d lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as $|r{|}^{-\rho }$, which is expected to feature the same critical behavior of the long range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetisation and of the Binder cumulant, we present numerical evidence that ${\rho }_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless (BKT) universality class for $\rho \ge 4$, and to present a $2^{nd}$-order transition for $\rho <4$.

arXiv comment28 pages, 9 figures

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