Vortex-loop scaling in the three-dimensionalXYferromagnet

A three-dimensional (3D) generalization of the two-dimensional Kosterlitz real-space scaling procedure yields scaling equations for the 3D XY vortex-loop fugacity $y_l$ and loop coupling $K_l$. These agree with the equations obtained by Williams from helium phenomenology. A dominant diverging loop diameter a==$e^l$=${\xi }_{\mathrm{-}}$≊‖ε${\Vert }^{\mathrm{-}\nu }$ [ε==(T-$T_c$)/$T_c$] controls the exponents for the helicity modulus ${\rho }_s$≊‖ε${\Vert }^{\nu }$, spin-spin correlation length ∝${\xi }_{\mathrm{-}}$, specific heat α=2-3ν, and magnetic field response γ=2ν, Δ=(5/2ν, δ=5. ν is found by linearizing about a fixed point. For a loop core size $a_c$ small compared to the diverging diameter, $a_c$/${\xi }_{\mathrm{-}}$→0, a ‘‘Gaussian’’ value ν=(1/2 results. An ansatz $a_c$(l)∝a is made to represent the onset of screening from crinkled loops: $a_c$(l)/a≃($K_l$ ${)}^{\theta }$ with θ the 3D self-avoiding walk exponent ≊0.6. This divergent-core ansatz [$a_c$(l-)/${\xi }_{\mathrm{-}}$=0.57] yields the exponents ν=0.67, γ=1.34, α=-0.015. Monte Carlo results of Kohring et al. on vorticity suppression are discussed.