Anisotropic three-dimensionalXYmodel 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 ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$=${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}2}$. For high-${\mathit{T}}_{\mathit{c}}$ superconductors, ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}2}$, related to the Ginzburg-Landau masses, is in a strong anisotropy regime, ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$0.5, and a length scale ${\mathit{r}}_{0}$=${\ensuremath{\gamma}}_{0}$${\mathit{a}}_{0}$ naturally arises (${\mathit{a}}_{0}$= lattice constant). (i) For loop major axes a>${\mathit{r}}_{0}$, the dominant excitations are 3D elliptical vortex loops cutting multiple planes, with a\ensuremath{\sim}${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$(T)=${\mathit{a}}_{0}$\ensuremath{\Vert}\ensuremath{\varepsilon}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ where \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}\ensuremath{\equiv}\ensuremath{\Vert}(T-${\mathit{T}}_{\mathit{c}}$)/${\mathit{T}}_{\mathit{c}}$\ensuremath{\Vert}, and \ensuremath{\nu}=0.67. The vorticity segment components (\ensuremath{\mu}=x^,y^,z^) are ${\mathit{J}}_{\mathrm{\ensuremath{\mu}}}$(r)=0,\ifmmode\pm\else\textpm\fi{}1, and interact via a Biot-Savart-like law. The renormalized anisotropy ${\ensuremath{\gamma}}_{\mathit{l}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}1, asymptotically isotropic, as l=ln(a/${\mathit{a}}_{0}$)\ensuremath{\rightarrow}\ensuremath{\infty}: anisotropy is irrelevant. (ii) For loop scales ${\mathit{r}}_{0}$>a>2${\mathit{a}}_{0}$ the dominant excitations are quasi-2D rectangular loops with short sides ${\mathit{J}}_{\mathit{z}}$(r)=\ifmmode\pm\else\textpm\fi{}1 cutting single planes, that are thus effectively decoupled at finite scales ${\mathit{r}}_{0}$. The ${\mathit{J}}_{\mathit{z}}$=\ifmmode\pm\else\textpm\fi{}1 vortex components, of in-plane separation \ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{W}}$mN\ensuremath{\Vert}${\mathit{r}}_{0}$, interact via a logarithmic [ln(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/${\mathit{r}}_{0}$)] plus linear [${\mathit{Q}}_{0}$(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/${\mathit{a}}_{0}$-1)] potential. As ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}0, the coefficient ${\mathit{Q}}_{0}$\ensuremath{\rightarrow}0, and the Kosterlitz-Thouless limit is recovered. (iii)The 3D transition temperature ${\mathit{T}}_{\mathit{c}}$ versus ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$, calculated for strong anisotropies from ``2D'' (3D) scaling for scales ${\mathit{r}}_{0}$ (>${\mathit{r}}_{0}$), matches existing Monte Carlo data well. Critical regions \ensuremath{\Vert}${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$\ensuremath{\Vert}\ensuremath{\le}${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1/\ensuremath{\nu}}$ are estimated. Contact is made with ideas of an intrinsic critical current arising from linear \ensuremath{\sim}${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$${\mathit{R}}_{\mathrm{\ensuremath{\parallel}}}$ vortex segment effective potentials.