Vortex-loop scaling in the three-dimensional XY ferromagnet

Abstract

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=ξ_≊||ε||^−ν [ε==(T-T_c)/T_c] controls the exponents for the helicity modulus ρ_s≊||ε||^ν, spin-spin correlation length ∝ξ_-, 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/ξ_-→0, a "Gaussian" value ν=(½ results. An ansatz a_c(l)∝a is made to represent the onset of screening from crinkled loops: a_c(l)/a≃(K_l)^θ with θ the 3D self-avoiding walk exponent ≊0.6. This divergent-core ansatz [a_c(l-)/ξ_-=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.