The anomalous dimension of the lattice London superconductor is determined from finite-size scaling of the susceptibility to be η ϕ = − 0.79(1). It is furthermore found that the vanishing of the vortex loop line tension in the 3D XY model, which should be related to that exponent, agrees reasonable well with this value of η ϕ . Deviations from this behavior are however also identified and it is suggested that they are an effect of the vortex loop intersections and therefore will vanish only in the limit of low vortex density.
The dynamic critical exponent z is determined from numerical simulations for the three-dimensional XY model subject to two types of dynamics, i.e. , relaxational dynamics and resistively shunted junction (RSJ) dynamics, as well as for two different treatments of the boundary, i.e. , periodic boundary condition (PBC) and fluctuating twist boundary condition (FTBC). In case of relaxational dynamics, finite-size scaling at the critical temperature gives z ≈ 2 for PBC and 1.5 for FTBC, while for RSJ dynamics z ≈ 1.5 is obtained in both cases. The results are discussed in the context of superfluid/superconductors and vortex dynamics, and are compared with what has been found for other related models.
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>2
a0 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.
The fractal structure of high-temperature graphs of the three-dimensional Ising and XY models is investigated by simulating these graphs directly on a cubic lattice and analyzing them with the help of percolation observables. The Ising graphs are shown to percolate right at the Curie critical point. The diverging length scale relevant to the graphs in the vicinity of the percolation threshold is shown to be provided by the spin correlation length. The fractal dimension of the high-temperature graphs at criticality is estimated to be D=1.7349(65) for the Ising and D=1.7626(66) for the XY models.
Hove, Mo, and Sudbo [Phys. Rev. Lett. 85, 2368 (2000)] derived a simple connection,
η+DH=2, between the anomalous scaling dimension
η of the U(1) universality class order parameter and the Hausdorff dimension
DH of critical loops in loop representations of U(1) models. We show that the above relation is wrong and establish a correct relation that contains a new critical exponent.
To analyze the
±J XY spin-glass in three dimensions, we verified a method aimed at obtaining a high-precision dynamical exponent
z from the correlation length in the nonequilibrium relaxation process. The obtained
z yielded consistent and highly accurate results in previous studies for relatively well-studied models---specifically, the three-dimensional (3D) ferromagnetic Ising model and the 3D
±J Ising model. Building on these previous studies, we used this method and the dynamical scaling method to analyze the 3D
±J XY model and obtained highly precise critical temperatures and exponents. The estimated transition temperatures satisfy $T_{\mathrm{SG}}