The role of screening in the vortex-loop phase transition in the anisotropic 3D X-Y model (describing HT c superconductors, etc.) is considered. RG analysis shows that effects of screening plays a decisive role in the 3D X-Y model contrary to the 2D one. As a result, a Kosterlitz-Thouless-like criterion of the phase transition, related with the appearance of a single vortex thread, which is disclosed on the system boundaries and has a smooth contour, cannot be formulated. Coulomb-type interactions between elements of the entangled loops deform and reclose these loops in order to screen the interactions. As a result one can consider vortex configurations corresponding to the effective medium with Debye-type screening. The polymer physics approach is used to show that parts of the contour of the entangled vortex loops undergo the tangle-globule transition. Due to the approach used a Kosterlitz-Thouless-like criterion for the phase transition may be formulated. As a result an estimation of the critical temperature of the phase transition is obtained for the 3D X-Y model with arbitrary anisotropy.
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-Tc 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}$0.5, and a length scale r0=${\ensuremath{\gamma}}_{0}$a0 naturally arises (a0= lattice constant). (i) For loop major axes a>r0, the dominant excitations are 3D elliptical vortex loops cutting multiple planes, with a\ensuremath{\sim}${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$(T)=a0\ensuremath{\Vert}\ensuremath{\varepsilon}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ where \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}\ensuremath{\equiv}\ensuremath{\Vert}(T-Tc)/Tc\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/a0)\ensuremath{\rightarrow}\ensuremath{\infty}: anisotropy is irrelevant. (ii) For loop scales r0>a>2a0 the dominant excitations are quasi-2D rectangular loops with short sides Jz(r)=\ifmmode\pm\else\textpm\fi{}1 cutting single planes, that are thus effectively decoupled at finite scales r0. The Jz=\ifmmode\pm\else\textpm\fi{}1 vortex components, of in-plane separation \ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{W}}$mN\ensuremath{\Vert}r0, interact via a logarithmic [ln(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/r0)] plus linear [Q0(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/a0-1)] potential. As ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}0, the coefficient Q0\ensuremath{\rightarrow}0, and the Kosterlitz-Thouless limit is recovered. (iii)The 3D transition temperature Tc versus ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$, calculated for strong anisotropies from ``2D'' (3D) scaling for scales r0 (>r0), 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.
Monte Carlo simulations of a 3D anisotropic Josephson junction array are performed, to determine the critical temperature T c ( I ,α) for the onset of phase incoherence. Here I is the external current drive at opposite surfaces, and α≡ E J ⊥ / E J // is the inter-/intra-layer Josephson coupling ratio. T c ( I ,α) is equivalent to a critical current, I c ( T ,α) for the breakdown of global phase coherence. The temperature and anisotropy dependence of I c ( T ,α) supports a physical picture of two vortex populations, quasi-2D and 3D in nature at small and large scales respectively, separated by a length scale r⊥∼1/α.