Anisotropic three-dimensional XY model and vortex-loop scaling

Abstract

Scaling equations are obtained for thermally excited vortex loops in the anisotropic three-dimensional (3D) XY model with interplane/intraplane coupling ratio K_⊥/K_||=γ₀⁻². For high-T_c superconductors, γ₀⁻², related to the Ginzburg-Landau masses, is in a strong anisotropy regime, γ₀⁻¹ < 0.5, and a length scale r₀=γ₀a₀ naturally arises (a₀= lattice constant). (i) For loop major axes a > r₀, the dominant excitations are 3D elliptical vortex loops cutting multiple planes, with a~ξ_(T)=a₀||ε||^−ν where ||ε||≡||(T-T_c)/T_c||, and ν=0.67. The vorticity segment components (μ=x^,y^,z^) are J_μ(r)=0,±1, and interact via a Biot-Savart-like law. The renormalized anisotropy γ_l⁻¹→1, asymptotically isotropic, as l=ln(a/a₀)→∞: anisotropy is irrelevant. (ii) For loop scales r₀ > a > 2a₀ the dominant excitations are quasi-2D rectangular loops with short sides J_z(r)=±1 cutting single planes, that are thus effectively decoupled at finite scales < r₀. The J_z=±1 vortex components, of in-plane separation ||R→_WmN|| < r₀, interact via a logarithmic [ln(||R→_||||/r₀)] plus linear [Q₀(||R→_||||/a₀-1)] potential. As γ₀⁻¹→0, the coefficient Q₀→0, and the Kosterlitz-Thouless limit is recovered. (iii) The 3D transition temperature T_c versus K_⊥/K_||, calculated for strong anisotropies from "2D" (3D) scaling for scales < r₀ (> r₀), matches existing Monte Carlo data well. Critical regions ||ε_c||=||≤γ₀^−1/ν are estimated. Contact is made with ideas of an intrinsic critical current arising from linear ~γ₀⁻¹R_|| vortex segment effective potentials.