The 2D XY model on a finite lattice with structural disorder: quasi-long-range ordering under realistic conditions

TL;DR

This paper analytically and numerically investigates how quenched disorder and finite size affect the quasi-long-range order in the 2D XY model, revealing a non-universal decay exponent influenced by temperature and impurity concentration.

Contribution

It introduces an analytic spin-wave approach combined with Monte Carlo simulations to study disorder effects on the 2D XY model's ordering properties.

Findings

The spin-spin correlation decay exponent depends on temperature and impurity concentration.

Magnetisation vanishes with increasing lattice size, consistent with the Mermin-Wagner-Hohenberg theorem.

Analytic results agree with Monte Carlo simulations across different impurity concentrations.

Abstract

We present an analytic approach to study concurrent influence of quenched non-magnetic site-dilution and finiteness of the lattice on the 2D XY model. Two significant deeply connected features of this spin model are: a special type of ordering (quasi-long-range order) below a certain temperature and a size-dependent mean value of magnetisation in the low-temperature phase that goes to zero (according to the Mermin-Wagner-Hohenberg theorem) in the thermodynamic limit. We focus our attention on the asymptotic behaviour of the spin-spin correlation function and the probability distribution of magnetisation. The analytic approach is based on the spin-wave approximation valid for the low-temperature regime and an expansion in the parameters which characterise the deviation from completely homogeneous configuration of impurities. We further support the analytic considerations by Monte Carlo simulations performed for different concentrations of impurities and compare analytic and MC results. We present as the main quantitative result of the work the exponent of the spin-spin correlation function power law decay. It is non universal depending not only on temperature as in the pure model but also on concentration of magnetic sites. This exponent characterises also the vanishing of magnetisation with increasing lattice size.