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arXiv:cond-mat/0003447 · Physical review. B, Condensed matter

Dynamic critical exponent of two-, three-, and four-dimensional XY models with relaxational and resistively shunted junction dynamics

Lars Melwyn Jensen, Beom Jun Kim, Petter Minnhagen

Published Jun 1, 2000• 44 citations
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Abstract

The dynamic critical exponent z is determined numerically for the d-dimensional $\mathrm{XY}$ model $(d=2,$ 3, and 4) subject to relaxational dynamics and resistively shunted junction dynamics. We investigate both the equilibrium fluctuation and the relaxation behavior from nonequilibrium towards equilibrium, using the finite-size scaling method. The resulting values of z are shown to depend on the boundary conditions used, the periodic boundary condition, and fluctuating twist boundary condition (FTBC), which implies that the different treatments of the boundary in some cases give rise to different critical dynamics. It is also found that the equilibrium scaling and the approach to equilibrium scaling for the same boundary condition do not always give the same value of z. The FTBC in conjunction with the finite-size scaling of the linear resistance for both type of dynamics yields values of z consistent with expectations for superfluids and superconductors: $z=2,$ 3/2, and 2 for $d=2,$ 3, and 4, respectively.

DOI
10.1103/physrevb.61.15412

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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.
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Two-dimensional XY\mathrm{XY}XY models with resistively shunted junction (RSJ) dynamics and time dependent Ginzburg-Landau (TDGL) dynamics are simulated and it is verified that the vortex response is well described by the Minnhagen phenomenology for both types of dynamics. Evidence is presented supporting that the dynamical critical exponent z in the low-temperature phase is given by the scaling prediction (expressed in terms of the Coulomb gas temperature TCG{T}^{\mathrm{CG}}TCG and the vortex renormalization given by the dielectric constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}) z=1/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}{T}^{\mathrm{CG}}\ensuremath{-}2>~2$ both for RSJ and TDGL and that the nonlinear IV\mathrm{IV}IV exponent a is given by a=z+1a=z+1a=z+1 in the low-temperature phase. The results are discussed and compared with the results of other recent papers and the importance of the boundary conditions is emphasized.
It is shown from computer simulations that the current-voltage $(I\ensuremath{-}V)$ characteristics for the two-dimensional XY model with resistively shunted Josephson junction dynamics and Monte Carlo dynamics obeys a finite-size scaling form from which the nonlinear $I\ensuremath{-}V$ exponent a can be determined to good precision. This determination supports the conclusion a=z+1,a=z+1,a=z+1, where z is the dynamic critical exponent. The results are discussed in the light of the contrary conclusion reached by Tang and Chen [Phys. Rev. B 67, 024508 (2003)] and the possibility of a breakdown of scaling suggested by Bormann [Phys. Rev. Lett. 78, 4324 (1997)].