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arXiv:cond-mat/0010121 · Physical review. B, Condensed matter

Current-voltage characteristics of the two-dimensional XY model with Monte Carlo dynamics

Beom Jun Kim

Published Dec 13, 2000• 10 citations
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Abstract

Current-voltage characteristics and the linear resistance of the two-dimensional $\mathrm{XY}$ model with and without external uniform current driving are studied by Monte Carlo simulations. We apply the standard finite-size scaling analysis to get the dynamic critical exponent z at various temperatures. From the comparison with the resistively shunted junction dynamics, it is concluded that z is universal in the sense that it does not depend on details of dynamics. This comparison also leads to the quantification of the time in the Monte Carlo dynamic simulation.

DOI
10.1103/physrevb.63.024503

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Current-voltage characteristics and the linear resistance of the two-dimensional XY\mathrm{XY}XY model with and without external uniform current driving are studied by Monte Carlo simulations. We apply the standard finite-size scaling analysis to get the dynamic critical exponent z at various temperatures. From the comparison with the resistively shunted junction dynamics, it is concluded that z is universal in the sense that it does not depend on details of dynamics. This comparison also leads to the quantification of the time in the Monte Carlo dynamic simulation.
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)].
Dynamical simulations and scaling arguments are used to study the current-voltage $(I\ensuremath{-}V)$ characteristics of a two-dimensional model of resistively shunted Josephson-junction arrays in presence of percolative disorder, at zero external field. Two different limits of the Josephson-coupling concentration ppp are considered, where pc{p}_{c}pc​ is the percolation threshold. For p>pcp>{p}_{c}p>pc​ and zero temperature, the $I\ensuremath{-}V$ curves show power-law behavior above a disorder-dependent critical current. The power-law behavior and critical exponents are consistent with a simple scaling analysis. At pc{p}_{c}pc​ and finite temperature T,T,T, the results show the scaling behavior of a T=0T=0T=0 superconducting transition. The resistance is linear but vanishes for decreasing TTT with an apparent exponential behavior. Crossover to nonlinearity appears at currents proportional to ${T}^{1+{\ensuremath{\nu}}_{T}},$ with a thermal-correlation length exponent ${\ensuremath{\nu}}_{T}$ consistent with the corresponding value for the diluted XY\mathrm{XY}XY model at pc.{p}_{c}.pc​.
It has been suggested [Pierson et al., Phys. Rev. B 60, 1309 (1999); Ammirata et al., Physica C 313, 225 (1999)] that for two-dimensional (2D) superconductors there exists a phase transition with the dynamic critical exponent $z\ensuremath{\approx}5.6.$ We perform simulations for the 2D resistively-shunted-junction model and compare the results with the experimental data in Repaci et al. obtained for an ultrathin YBCO sample [Phys. Rev. B 54, R9674 (1996)]. We then use a different method of analyzing dynamic scaling than in Pierson et al., and conclude that both the simulations and the experiments are consistent with a conventional Kosterlitz Thouless (KT) transition in the thermodynamic limit for which z=2.z=2.z=2. For finite systems, however, we find both in simulations and experiments that the change in the current-voltage $(I\ensuremath{-}V)$ characteristics caused by the finite size shows a scaling property with an exponent $\ensuremath{\alpha}\ensuremath{\approx}1/6,$ seemingly suggesting a vanishing resistance at a temperature for which $z={\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is pointed out that the dynamic critical exponent found in Pierson et al. corresponds to the exponent ${\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is emphasized that this scaling property does not represent any true phase transition since in reality the resistance vanishes only at zero temperature. Nevertheless, the observed scaling behavior associated with $\ensuremath{\alpha}\ensuremath{\approx}1/6$ appears to be a common and intriguing feature for the finite size caused change in the $I\ensuremath{-}V$ characteristics around the KT transition.
We numerically study dynamic critical behavior of the one-dimensional XYXYXY model with a long-range interaction by using the Monte Carlo method and the resistively-shunted Josephson junction model. The two dynamic models exhibit the mean-field universality class in equilibrium as expected, but the dynamic critical behavior is shown to sensitively depend on details of numerical simulation. In more detail, the trial angle range in the Monte Carlo simulation is found to alter the value of the dynamic critical exponent, and the scaling of the Monte Carlo time unit by the acceptance ratio is shown to be useful to improve the estimation of the dynamic critical exponent. We compare the Monte Carlo result of the dynamic critical exponent with the result from the more realistic dynamic model of the resistively-shunted junction. We conclude that the small value of the trial angle range should be used to properly detect dynamic critical behavior.
Short-time dynamic behaviour of two-dimensional fully frustrated XY model near the Ising-like phase-transition temperature is studied with resistively shunted junction dynamics. The transition temperature T c and the dynamic and static critical exponents z , 2β/ν and ν are estimated under both a periodic boundary condition (PBC) and a fluctuating twist boundary condition (FTBC) using the short-time dynamic scaling analysis. The results show that T c , 2β/ν and ν are almost independent of the boundary conditions, whereas z depends on the boundary condition. The value of z = 2.18(5) of the PBC is roughly equal to that of the FF XY model with Monte Carlo dynamics under the PBC, while z = 2.05(4) of the FTBC is in agreement with that of the Coulomb gas model.
The dynamic critical exponent z is determined numerically for the d-dimensional XY\mathrm{XY}XY model (d=2,(d=2,(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,z=2,z=2, 3/2, and 2 for d=2,d=2,d=2, 3, and 4, respectively.