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Physical Review Letters

High-Precision Evaluation of the Static Exponents of the Classical Heisenberg Ferromagnet

Robert Goodell Brown, Mikael Ciftan

Published Feb 19, 1996• 23 citations
Publisher

Abstract

A high-precision, multimethod Monte Carlo study of the static exponents of the $O(3)$ classical Heisenberg ferromagnet yields an estimate for the critical temperature of ${T}_{c}\ensuremath{\approx}1.4427\ifmmode\pm\else\textpm\fi{}0.0002$ from which it is conjectured that ${T}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/\mathrm{ln}(2)$. It is shown from finite-size scaling of the data that the specific heat is singular for this model, with $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.0750\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.642\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.328\ifmmode\pm\else\textpm\fi{}0.001$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.270\ifmmode\pm\else\textpm\fi{}0.001$. This result is confirmed by comparing the universality scaling of the full order parameter curve for three possible values of $\ensuremath{\nu}$.

DOI
10.1103/physrevlett.76.1352

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A high-precision, multimethod Monte Carlo study of the static exponents of the O(3)O(3)O(3) classical Heisenberg ferromagnet yields an estimate for the critical temperature of Tc\ensuremath≈1.4427\ifmmode±\else\textpm\fi0.0002{T}_{c}\ensuremath{\approx}1.4427\ifmmode\pm\else\textpm\fi{}0.0002Tc​\ensuremath≈1.4427\ifmmode±\else\textpm\fi0.0002 from which it is conjectured that Tc=1/ln(2){T}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/\mathrm{ln}(2)Tc​=1/ln(2). It is shown from finite-size scaling of the data that the specific heat is singular for this model, with \ensuremathα=0.0750\ifmmode±\else\textpm\fi0.002\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.0750\ifmmode\pm\else\textpm\fi{}0.002\ensuremathα=0.0750\ifmmode±\else\textpm\fi0.002, \ensuremathν=0.642\ifmmode±\else\textpm\fi0.002\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.642\ifmmode\pm\else\textpm\fi{}0.002\ensuremathν=0.642\ifmmode±\else\textpm\fi0.002, \ensuremathβ=0.328\ifmmode±\else\textpm\fi0.001\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.328\ifmmode\pm\else\textpm\fi{}0.001\ensuremathβ=0.328\ifmmode±\else\textpm\fi0.001, and \ensuremathγ=1.270\ifmmode±\else\textpm\fi0.001\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.270\ifmmode\pm\else\textpm\fi{}0.001\ensuremathγ=1.270\ifmmode±\else\textpm\fi0.001. This result is confirmed by comparing the universality scaling of the full order parameter curve for three possible values of \ensuremathν\ensuremath{\nu}\ensuremathν.
Using extensive Monte Carlo simulations, we study the equilibrium properties of the simple-cubic, classical Heisenberg ferromagnet. We employ very long runs for L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices to obtain high-precision data for the magnetization probability distribution. Using finite-size scaling for L\ensuremath{\le}24 and an optimized multiple-histogram data analysis, we obtain an accurate value of the inverse critical temperature J/kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​=0.6929\ifmmode\pm\else\textpm\fi{}0.0001, which is higher than previously accepted estimates. Calculated values of various static exponents are in excellent agreement with renormalization-group and \ensuremath{\epsilon}-expansion predictions.
Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
In a recent letter, R.G. Brown and M. Ciftan (Phys. Rev. Lett. 76, 1352, 1996) reported high precision Monte Carlo (MC) estimates of the static critical exponents of the classical 3D Heisenberg model, which stand in sharp contrast to values obtained by four independent approaches, namely by other recent high statistics MC simulations, high-temperature series analyses, field theoretical methods, and experimental studies. In reply to the above cited work we submitted this paper as a comment to Phys. Rev. Lett.
The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.
Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$, HJ< 0.5\frac{H}{J}<~0.5JH​< 0.5 the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.
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.