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Physical Review A

One-dimensional critical dynamics and conservation laws

J. Kamphorst Leal da Silva, F. C. Sá Barreto

Published Aug 1, 1991• 3 citations
Publisher

Abstract

The one-dimensional kinetic Ising models with order-parameter conservation are studied for different transition rates. The effect on critical dynamics of global and local conservation laws is considered. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown for several dynamical models that z\ensuremath{\ge}5 even if the order-parameter conservation occurs only globally. For models in which the maximal separation between two exchanged spins is proportional to the equilibrium correlation length, the inequality z\ensuremath{\ge}3 is obtained. The d-dimensional case is discussed briefly.

DOI
10.1103/physreva.44.2727

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The one-dimensional kinetic Ising models with order-parameter conservation are studied for different transition rates. The effect on critical dynamics of global and local conservation laws is considered. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown for several dynamical models that z\ensuremath{\ge}5 even if the order-parameter conservation occurs only globally. For models in which the maximal separation between two exchanged spins is proportional to the equilibrium correlation length, the inequality z\ensuremath{\ge}3 is obtained. The d-dimensional case is discussed briefly.
One-dimensional kinetic Ising models with long-range exchange of two spins with opposite signs are considered. The exchange probability is a power-law function of the distance between exchanged spins. Rigorous lower bounds for the dynamical exponent z are obtained from the initial response and the scaling hypothesis of the relaxation time. The dependence of the z exponent on the maximal exchange distance and on the exchange probability is studied. Nonuniversal behavior in the ferromagnetic model with two different alternating coupling constants is found. The z exponent is also obtained from the domain-wall argument.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.
The one-dimensional kinetic Potts models are studied for both Glauber and Kawasaki dynamics with different transition rates. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown that the z exponent is non-universal for several periodic models with a basic unit cell (J 1 ,J 2 ,. . .,JM) containing different coupling constants. The critical dynamics of the alternating bond model is also studied by a simple argument about the movement of domain walls.
We extend a previous Monte Carlo investigation to include the relaxation times associated with the first time moment of the relaxation function and of the relaxation function squared. Moreover, we introduce, in addition to the order-parameter and energy-relaxation functions, that of the order parameter squared. The asymptotic temperature dependence of the associated relaxation times is found to support the validity of rather extended dynamic scaling hypotheses, predicting that all the associated exponents of slowing down are equal and symmetric with respect to Tc{T}_{c}Tc​. We also derive various exact lower bounds and inequalities for these relaxation times and exponents. Our numerical data are found to be consistent with these exact results.