Vortex-loop crinkling in the three-dimensionalXYmodel: Numerical evidence in support of an ansatz

A scaling theory for the three-dimensional (3D) XY ferromagnet was previously proposed, in terms of directed vortex loops of average diameter a=eˡ, and core size ac(l). An ansatz was made for the loop self-energy from a 1/R potential, E{~}a ln[a/ac(l)]{→}a lnKₗ^-x, where Kₗ 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 ac(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 ac(l) is given.