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Journal of Applied Physics

Monte Carlo study of critical relaxation in the 3D Heisenberg model

Pawel Peczak, David Paul Landau

Published May 1, 1990• 33 citations
Publisher

Abstract

Employing Monte Carlo simulations, the equilibrium dynamic critical behavior or L×L×L simple cubic classical Heisenberg models with periodic boundary conditions are determined. From the long-time exponential decay of the time-displaced correlation function of the magnetization the linear (equilibrium) relaxation time τ is also determined. It is found that at the critical point Tc, τ grows like τ≂Lz with z=1.94±0.06. In the paramagnetic region a power-law divergence τ≂(T/Tc−1)−Δ was found with Δ=zν=1.38±0.04, which implies a dynamical critical exponent z=1.96±0.07. Both values are consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving forces.

DOI
10.1063/1.344578

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Employing Monte Carlo simulations, the equilibrium dynamic critical behavior or L×L×L simple cubic classical Heisenberg models with periodic boundary conditions are determined. From the long-time exponential decay of the time-displaced correlation function of the magnetization the linear (equilibrium) relaxation time τ is also determined. It is found that at the critical point Tc, τ grows like τ≂Lz with z=1.94±0.06. In the paramagnetic region a power-law divergence τ≂(T/Tc−1)−Δ was found with Δ=zν=1.38±0.04, which implies a dynamical critical exponent z=1.96±0.07. Both values are consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving forces.
We have carried out extensive Monte Carlo simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic classical Heisenberg models with periodic boundary conditions. The equilibrium relaxation time \ensuremath{\tau} is determined from the long-time exponential decay of the time-displaced correlation function of the magnetization. Using finite-size scaling and corrections to scaling we find that at the critical point Tc{\mathit{T}}_{\mathit{c}}Tc​=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}Lz{\mathit{L}}^{\mathit{z}}Lz with z=1.96(6). In the paramagnetic region we show that the relaxation time is represented by a universal scaling function \ensuremath{\tau}${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{z}}$ vs (T/Tc{\mathit{T}}_{\mathit{c}}Tc​-1)${\mathit{L}}^{1/\ensuremath{\nu}}$ with z=1.96(6). This estimate is consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving dipolar interactions. We also describe how a simple over-relaxation algorithm of Creutz and of Brown and Woch can be incorporated into the Metropolis sampling method for the classical Heisenberg model, thereby significantly reducing the critical slowing down; the resultant correlation time varies as \ensuremath{\tau}\ensuremath{\sim}(T/Tc{\mathit{T}}_{\mathit{c}}Tc​-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.
The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τL are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τL with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.