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arXiv:cond-mat/0103345 · Europhysics Letters (EPL)

Critical exponent η phi of the lattice London superconductor and vortex loops in the 3D XY model

Peter Olsson

Published Jun 1, 2002• 8 citations
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Abstract

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.

DOI
10.1209/epl/i2002-00406-6

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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 geometric properties of critical fluctuations in the three-dimensional (3D) XYXYXY model are analyzed. The 3D XYXYXY model is a lattice model describing superfluids. We present a direct evaluation of the Hausdorff dimension DH{D}_{H}DH​ of the vortex loops which are the critical fluctuations of the 3D XYXYXY model. We also present analytical arguments for why $\ensuremath{\vartheta}$ in the scaling relation ${\ensuremath{\eta}}_{\ensuremath{\phi}}+{D}_{H}=2+\ensuremath{\vartheta}$ between DH{D}_{H}DH​ and the anomalous scaling dimension of the corresponding field theory must be zero.
We have studied the linear resistance of a three dimensional lattice superconductor model in the London limit by Monte Carlo simulation of the vortex loop dynamics. We find excellent finite size scaling at the phase transition. We determine the dynamical exponent z=1.51z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.51z=1.51 for the isotropic London lattice model.
We carry out Monte Carlo simulations of the three dimensional (3D) lattice London superconductor in zero applied magnetic field, making a detailed finite size scaling analysis of the Meissner transition. We find that the magnetic penetration length $\ensuremath{\lambda}$ and the correlation length $\ensuremath{\xi}$ scale as $\ensuremath{\lambda}\ensuremath{\sim}\ensuremath{\xi}\ensuremath{\sim}|t{|}^{\ensuremath{-}\ensuremath{\nu}}$, with $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.66\ifmmode\pm\else\textpm\fi{}0.03$, consistent with ordinary 3D XY\mathrm{XY}XY universality, ${\ensuremath{\nu}}_{\mathrm{XY}}\ensuremath{\approx}2/3$. Our results confirm the anomalous scaling dimension of magnetic field correlations at Tc{T}_{c}Tc​.
Monte Carlo simulations have been used to study a version of the XY model without vortex loops for J/T=0 on simple cubic and face-centered-cubic lattices. Finite-size scaling of data from L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with L up to 32 is used to obtain values for the order parameter \ensuremath{\Vert}M\ensuremath{\Vert} and a 1/L finite-size correction on each lattice. The order-parameter susceptibility is found to be diverging as L0.8{\mathit{L}}^{0.8}L0.8. We then study the crossover from three-dimensional to one-dimensional behavior on L\ifmmode\times\else\texttimes\fi{}W\ifmmode\times\else\texttimes\fi{}W simple cubic lattices, with W=4 and 8, and L up to 1024.
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.
We have tested the leading correction-to-scaling exponent omega in O(n)-symmetric models on a three-dimensional lattice by analysing the recent Monte Carlo (MC) data. We have found that the effective critical exponent, estimated at finite sizes of the system L and L/2, decreases remarkably within the range of the simulated L values. This shows the incorrectness of some claims that omega has a very accurate value 0.845(10) at n=1. A selfconsistent infinite volume extrapolation yields row estimates omega=0.547, omega=0.573, and omega=0.625 at n=1, 2, and 3, respectively, in approximate agreement with the corresponding exact values 1/2, 5/9, and 3/5 predicted by our recently developed GFD (grouping of Feynman diagrams) theory. We have fitted the MC data for the susceptibility of 3D Ising model at criticality showing that the effective critical exponent eta tends to increase well above the usually accepted values around 0.036. We have fitted the data within [L;8L], including several terms in the asymptotic expansion with fixed exponents, to obtain the effective amplitudes depending on L. This method clearly demonstrates that the critical exponents of GFD theory are correct (the amplitudes converge to certain asymptotic values at L tending to infinity), whereas those of the perturbative renormalization group (RG) theory are incorrect (the amplitudes diverge). A modification of the standard Ising model by introducing suitable "improved" action (Hamiltonian) does not solve the problem in favour of the perturbative RG theory.