Vortex-loop crinkling in the three-dimensional XY model: numerical evidence in support of an ansatz

Abstract

A scaling theory for the three-dimensional (3D) XY ferromagnet was previously proposed, in terms of directed vortex loops of average diameter a=e^l, and core size a_c(l). An ansatz was made for the loop self-energy from a 1/R potential, E~a ln[a/a_c(l)]→a lnK^−x_l, where K_l is the screened coupling and x≃=0.6 the self-avoiding random-walk exponent. Here, the central role of the core-size ansatz in producing known 3D XY exponents is brought out in numerical solutions of the scaling equations. We then relate the cutoff a_c(l) to radial fluctuations around the mean radius a/2, and provide partial numerical support for the ansatz by computer-generated vortex-loop configurations. A Flory-type polymer argument for the core size a_c(l) is given.