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Journal of Statistical Mechanics Theory and Experiment

Short-time critical and coarsening dynamics of the classical three-dimensional Heisenberg model

Henrique A. Fernandes, Roberto da Silva, J. R. Drugowich de Felício

Published Oct 5, 2006• 17 citations

Abstract

The dynamic critical behaviour of the three-dimensional classical Heisenberg model was studied by using short-time Monte Carlo simulations. The dynamic critical exponents z , θ and θ g , as well as the static critical exponents ν and β, were estimated by following the time evolution of the z -component of the total magnetization and its moments of higher order. The anomalous dimension of the magnetization x 0 was also calculated from a relation obtained through finite size scaling in the non-equilibrium stage. Afterwards, the local persistence exponent θ l was obtained by following the power law behaviour of the local persistence probability when the system was quenched from high temperatures to the null one ( T = 0). The time evolution of P l ( t ) was also analysed for T >0 and the expected exponential stretch was observed.

DOI: 10.1088/1742-5468/2006/10/p10002

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The dynamic critical behaviour of the three-dimensional classical Heisenberg model was studied by using short-time Monte Carlo simulations. The dynamic critical exponents z , θ and θ g , as well as the static critical exponents ν and β, were estimated by following the time evolution of the z -component of the total magnetization and its moments of higher order. The anomalous dimension of the magnetization x 0 was also calculated from a relation obtained through finite size scaling in the non-equilibrium stage. Afterwards, the local persistence exponent θ l was obtained by following the power law behaviour of the local persistence probability when the system was quenched from high temperatures to the null one ( T = 0). The time evolution of P l ( t ) was also analysed for T >0 and the expected exponential stretch was observed.

Comprehensive Monte Carlo simulations of the short-time dynamic behaviour are reported for the three-dimensional Ising model at criticality. Besides the exponent of the critical initial increase and the dynamic exponent z , the static critical exponents and as well as the critical temperature are determined from the power-law scaling behaviour of observables at the beginning of the time evolution. States of very high temperature as well as of zero temperature are used as initial states for the simulations.

We have investigated the non-equilibrium critical behavior of the three-dimensional classical Heisenberg model with evolution from various initial states. Initial states with magnetization m 0 ≠ 0 lead to a new time scale with the exponent k > 0 which produces complex universal scaling forms on the temporal behavior of the magnetization, the autocorrelation function, and the response function. Aging effects have been revealed in two-time behavior of the autocorrelation function with evolution from various initial states. We show that the slowing down of the autocorrelation function during the evolution from high-temperature initial state is considerably more slow than for case of the evolution from low-temperature initial state. Using scaling forms of the magnetization and the autocorrelation function, the values of critical exponents z = 2.035(4), θ ' = 0.490(1), and k = 1/( θ ' + β / νz ) = 1.340(4) were calculated.

We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to

{100}^{3}

spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.

In this work we study the non-equilibrium properties of Heisenberg ferromagnetic films using Monte Carlo simulations by short-time dynamic method. By exploring the short-time scaling dynamics, we have found thickness dependency of critical exponents z, θ′ and β/v for ferromagnetic thin film. For calculating the critical exponents of ferromagnetic films we considered systems with linear size L = 128 and layers number N = 2; 4; 6; 10. Starting from initial configurations, the system was updated with Metropolis algorithm at the critical temperatures