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

## TL;DR

This study investigates the critical behavior of the 2D long-range diluted XY model, providing numerical evidence that the short-range threshold is at 4, which influences the universality class of the phase transition.

## Contribution

The paper determines the short-range threshold 4 for the 2D long-range diluted XY model using Monte Carlo simulations, clarifying its critical behavior and universality class.

## Key findings

- 4 is the threshold for the short-range behavior.
- The model exhibits BKT transition for 4 4.
- A second-order transition occurs for 4<4.

## 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, $\rho>\rho_{\rm sr}$, 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 $\rho_{\rm sr}$ 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 $\rho_{\rm sr}=4$. 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 $\rho<4$.