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arXiv:1906.04518 · Physical review. E

Computation of the dynamic critical exponent of the three-dimensional Heisenberg model

Antonio Astillero, Juan Jesus Ruiz-Lorenzo

Published Dec 16, 2019• 8 citations
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Abstract

Working in and out of equilibrium and using state-of-the-art techniques we have computed the dynamic critical exponent of the three-dimensional Heisenberg model. By computing the integrated autocorrelation time at equilibrium, for lattice sizes L≤64, we have obtained z=2.033(5). In the out-of-equilibrium regime we have run very large lattices (L≤250) obtaining z=2.04(2) from the growth of the correlation length. We compare our values with that previously computed at equilibrium with relatively small lattices (L≤24), with that provided by means a three-loops calculation using perturbation theory and with experiments. Finally we have checked previous estimates of the static critical exponents, η and ν, in the out-of-equilibrium regime.

DOI
10.1103/physreve.100.062117

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Working in and out of equilibrium and using state-of-the-art techniques we have computed the dynamic critical exponent of the three-dimensional Heisenberg model. By computing the integrated autocorrelation time at equilibrium, for lattice sizes L≤64, we have obtained z=2.033(5). In the out-of-equilibrium regime we have run very large lattices (L≤250) obtaining z=2.04(2) from the growth of the correlation length. We compare our values with that previously computed at equilibrium with relatively small lattices (L≤24), with that provided by means a three-loops calculation using perturbation theory and with experiments. Finally we have checked previous estimates of the static critical exponents, η and ν, in the out-of-equilibrium regime.
Working in the out of equilibrium regime and using state-of-the-art techniques we have computed the dynamic critical exponent of the three dimensional Heisenberg model. We have run very large lattices (L≤250L\le 250L≤250) in CPUs and GPUs obtaining z=2.041(16)z=2.041(16)z=2.041(16) from the growth of the correlation length and z=2.034(22)z=2.034(22)z=2.034(22) for the decay of the energy. We compare our values with that previously computed at equilibrium with relatively small lattices (L≤24L\le 24L≤24), with that provided by means a three-loops calculation using perturbation theory and with experiments. Finally we have checked previous estimates of the static critical exponents, η\etaη and ν\nuν, in this out of equilibrium regime.
We study the critical dynamics of the three-dimensional Heisenberg model with random cubic anisotropy in the out-of-equilibrium and equilibrium regimes. Analytical approaches based on field theory predict that the universality class of this model is that of the three-dimensional site-diluted Ising model. We have been able to estimate the dynamic critical exponent by working in the equilibrium regime and by computing the integrated autocorrelation times obtaining z=2.50(5) (without taking into account scaling corrections) and z=2.29(11) (by fixing the scaling corrections to that predicted by field theory). In the out-of-equilibrium regime we have focused on the study of the dynamic correlation length, allowing us to compute the dynamic critical exponent obtaining z=2.38(2), which is compatible with the equilibrium ones. Finally, both estimates are also compatible with the most accurate prediction z=2.35(2), from numerical simulations of the 3D site-diluted Ising model, in agreement with the predictions based on field theory.
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 1003{100}^{3}1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.
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.
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 improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find $\ensuremath{\gamma}=1.3960(9),$ $\ensuremath{\nu}=0.7112(5),$ $\ensuremath{\eta}=0.0375(5),$ $\ensuremath{\alpha}=\ensuremath{-}0.1336(15),$ $\ensuremath{\beta}=0.3689(3),$ and $\ensuremath{\delta}=4.783(3).$ We consider an improved lattice ${\ensuremath{\varphi}}^{4}$ Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the ${\ensuremath{\varphi}}^{4}$ improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.