Vortex-loop crinkling in the three-dimensional<i>XY</i>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=${\mathit{e}}^{\mathit{l}}$, and core size ${\mathit{a}}_{\mathit{c}}$(l). An ansatz was made for the loop self-energy from a 1/R potential, E\ensuremath{\sim}a ln[a/${\mathit{a}}_{\mathit{c}}$(l)]\ensuremath{\rightarrow}a ln${\mathit{K}}_{\mathit{l}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, where ${\mathit{K}}_{\mathit{l}}$ is the screened coupling and x\ensuremath{\simeq}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 ${\mathit{a}}_{\mathit{c}}$(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 ${\mathit{a}}_{\mathit{c}}$(l) is given.