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

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 $η=η(T/J)$ ($J$ 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 $\partialη/\partial p$ at point $p=0$, one can predict the value of the exponent for small dilution concentrations: $η(p)\simeqη(0)+p(\partialη/\partial p)|_{p=0}$. As it is shown in this paper, the spin-wave Hamiltonian allows to obtain exact results for this slope: $(\partialη/\partial p)|_{p=0} = T/(2J) + O((T/J)^2)$ and $T/(πJ) + O((T/J)^2)$ for site and for bond dilution, respectively.