Magnetic Correlation Function of the Two-Dimensional Planar Spin Model: Self-Consistent Calculation

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Abstract
The magnetic correlation function g(r) of the Villain model is calculated by applying the dual transformation to the discrete Gaussian (DG) model and by approximating it by the modified discrete Gaussian (mDG) model. g(r) decays with a power law with an exponent η below the critical temperature Tc, which corresponds to the roughening point in the mDG model. The susceptibility χ is infinite below Tc. Above Tc, g(r) decays exponentially with a magnetic correlation length ξM, which behaves similarly to the height correlation length ξ in mDG model: ln ξM ∼(1-Tc/T)-∼̅ν̅ with the exponent ∼̅ν̅=1. The susceptibility χ is proportional to ξ2- η(T)M, where the exponent η takes the value 1/4 at Tc. The value η(Tc)=1/4 agrees with that by Kosterlitz or the experimental results, although the value ∼̅ν̅=1 is different from ∼̅ν̅=1/2 obtained by Kosterlitz. Step free energy and the height difference correlation function in the mDG model are also calculated.