Critical behavior of the XY model in complex topologies

TL;DR

This paper investigates the critical behavior of the XY model on complex Levy graph topologies, revealing how the long-range connectivity influences phase transitions and effective dimensions through extensive GPU-based Monte Carlo simulations.

Contribution

It provides new insights into the relationship between long-range connectivity exponents and effective dimensions in the XY model on Levy graphs, using large-scale numerical simulations.

Findings

Mean-field and non-mean-field regimes identified

Effective short-range dimension matches spectral dimension in mean-field regime

Critical behavior depends on the Levy graph's connectivity exponent

Abstract

The critical behavior of the O(2) model on dilute Levy graphs built on a 2D square lattice is analyzed. Different qualitative cases are probed, varying the exponent rho governing the dependence on the distance of the connectivity probability distribution. The mean-field regime, as well as the long-range and short-range non-mean-field regimes are investigated by means of high-performance parallel Monte-Carlo numerical simulations running on GPUs. The relationship between the long-range rho exponent and the effective dimension of an equivalent short-range system with the same critical behavior is investigated. Evidence is provided for the effective short-range dimension to coincide with the spectral dimension of the Levy graph for the XY model in the mean-field regime.