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arXiv:hep-lat/9605024 · Physical Review Letters

Critical Exponents of the Classical Heisenberg Ferromagnet

Christian Holm, Wolfhard Janke E

Published Mar 17, 1997• 18 citations
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Abstract

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.

DOI
10.1103/physrevlett.78.2265

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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.
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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 ${T}_{c}\ensuremath{\approx}1.4427\ifmmode\pm\else\textpm\fi{}0.0002$ 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 $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.0750\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.642\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.328\ifmmode\pm\else\textpm\fi{}0.001$, 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 $\ensuremath{\nu}$.
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.
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.
Monte Carlo simulations near the phase transition. We compute high precision estimates of the critical coupling Kc, Binder’s parameter U ∗ , and the critical exponents ν, β/ν, η, and α/ν, using extensively histogram reweighting and optimization techniques that allow us to keep control over the statistical errors. Measurements of the autocorrelation time show the expected reduction of critical slowing down at the phase transition as compared to local update algorithms. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors in finite-size scaling analyses. 1.
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.