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arXiv:cond-mat/0301510 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dynamic critical behavior of the XY model in small-world networks

Kateryna Medvedyeva, Petter Holme, Petter Minnhagen, Beom Jun Kim

Published Mar 24, 2003• 59 citations
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Abstract

The critical behavior of the XY model on small-world network is investigated by means of dynamic Monte Carlo simulations. We use the short-time relaxation scheme, i.e., the critical behavior is studied from the nonequilibrium relaxation to equilibrium. Static and dynamic critical exponents are extracted through the use of the dynamic finite-size scaling analysis. It is concluded that the dynamic universality class at the transition is of the mean-field nature. We also confirm numerically that the value of dynamic critical exponent is independent of the rewiring probability P for P greater, similar 0.03.

DOI
10.1103/physreve.67.036118

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The critical behavior of the XY model on small-world network is investigated by means of dynamic Monte Carlo simulations. We use the short-time relaxation scheme, i.e., the critical behavior is studied from the nonequilibrium relaxation to equilibrium. Static and dynamic critical exponents are extracted through the use of the dynamic finite-size scaling analysis. It is concluded that the dynamic universality class at the transition is of the mean-field nature. We also confirm numerically that the value of dynamic critical exponent is independent of the rewiring probability P for P greater, similar 0.03.
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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.
Monte Carlo simulations of the short-time critical dynamics are reviewed. The short-time universal scaling behavior of the dynamic Ising model and Potts model are discussed in detail, while extension and application to more complex systems as the XY model, the fully frustrated XY model and other dynamic systems are also presented. The investigation of the universal behavior of the short-time dynamics not only enlarges the fundamental knowledge on critical phenomena but also, more interestingly, provides possible new ways to determine not only the new critical exponents θ and θ1, but also the traditional dynamic critical exponent z as well as all static critical exponents.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.