Spin-spin correlation function of the 2D XY model with weak site or bond dilution

TL;DR

This paper derives exact formulas for how weak site or bond dilution affects the decay exponent of the spin-spin correlation function in the 2D XY model below the BKT transition.

Contribution

It provides the first exact analytical results for the initial slope of the correlation exponent's dependence on dilution concentration.

Findings

Exact expressions for the slope of the exponent at zero dilution for site and bond dilution.

The slope depends linearly on temperature, with different coefficients for site and bond dilution.

Results enable prediction of the correlation decay exponent at small dilution levels.

Abstract

The spin-spin correlation function of the 2D XY model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature dependent exponent $\eta=\eta(T/J)$ ($J$ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D XY model with site or bond dilution this exponent depends on concentration $p$ of removed sites/bonds as well. Knowing the slope $\partial\eta/\partial p$ at point $p=0$, one can predict the value of the exponent for small dilution concentrations: $\eta(p)\simeq\eta(0)+p(\partial\eta/\partial p)|_{p=0}$. As it is shown in this paper, the spin-wave Hamiltonian allows to obtain exact results for this slope: $(\partial\eta/\partial p)|_{p=0} = T/(2J) + O((T/J)^2)$ and $T/(\pi J) + O((T/J)^2)$ for site and for bond dilution, respectively.