We use single-cluster Monte Carlo simulations to study the role of topological defects in the three-dimensional classical Heisenberg model on simple cubic lattices of size up to 80 3. By applying reweighting techniques to time series generated in the vicinity of the approximate infinite volume transition point Kc, we obtain clear evidence that the temperature derivative of the average defect density d〈n〉/dT behaves qualitatively like the specific heat, i.e., both observables are finite in the infinite volume limit. This is in contrast to results by Lau and Dasgupta [Phys. Rev. B39 (1989) 7212] who extrapolated a divergent behavior of d〈n〉/dT at Kc from simulations on lattices of size up to 16 3. We obtain weak evidence that d〈n〉/dT scales with the same critical exponent as the specific heat. As a byproduct of our simulations, we obtain a very accurate estimate for the ratio α/ν of the specific-heat exponent with the correlation-length exponent from a finite-size scaling analysis of the energy.We use single-cluster Monte Carlo simulations to study the role of topological defects in the three-dimensional classical Heisenberg model on simple cubic lattices of size up to 80 3. By applying reweighting techniques to time series generated in the vicinity of the approximate infinite volume transition point Kc, we obtain clear evidence that the temperature derivative of the average defect density d〈n〉/dT behaves qualitatively like the specific heat, i.e., both observables are finite in the infinite volume limit. This is in contrast to results by Lau and Dasgupta [Phys. Rev. B39 (1989) 7212] who extrapolated a divergent behavior of d〈n〉/dT at Kc from simulations on lattices of size up to 16 3. We obtain weak evidence that d〈n〉/dT scales with the same critical exponent as the specific heat. As a byproduct of our simulations, we obtain a very accurate estimate for the ratio α/ν of the specific-heat exponent with the correlation-length exponent from a finite-size scaling analysis of the energy.

Jan 1, 199419 citations

HIGH-PRECISION MONTE CARLO DETERMINATION OF α/ν IN THE 3D CLASSICAL HEISENBERG MODEL

Christian Holm, Wolfhard Janke E

To study the role of topological defects in the three-dimensional classical Heisenberg model we have simulated this model on simple cubic lattices of size up to 803, using the single-cluster Monte Carlo update. Analysing the specific-heat data of these simulations, we obtain a very accurate estimate for the ratio of the specific-heat exponent with the correlation-length exponent, α/ν, from a usual finite-size scaling analysis at the critical coupling Kc. Moreover, by fitting the energy at Kc, we reduce the error estimates by another factor of two, and get a value of α/ν, which is comparable in accuracy to best field theoretic estimates.To study the role of topological defects in the three-dimensional classical Heisenberg model we have simulated this model on simple cubic lattices of size up to 803, using the single-cluster Monte Carlo update. Analysing the specific-heat data of these simulations, we obtain a very accurate estimate for the ratio of the specific-heat exponent with the correlation-length exponent, α/ν, from a usual finite-size scaling analysis at the critical coupling Kc. Moreover, by fitting the energy at Kc, we reduce the error estimates by another factor of two, and get a value of α/ν, which is comparable in accuracy to best field theoretic estimates.

International Journal of Modern Physics CApr 1, 19940 citations

Critical exponents of the classical three-dimensional Heisenberg model: A single-cluster Monte Carlo study

Christian Holm, Wolfhard Janke E

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.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.

arXiv:hep-lat/9301002Physical review. B, Condensed matterJul 1, 1993232 citations

Ising model on three-dimensional random lattices: A Monte Carlo study

Wolfhard Janke E, Ramon Villanova

We report single-cluster Monte Carlo simulations of the Ising model on three-dimensional Poissonian random lattices with up to $128000\ensuremath{\approx}{50}^{3}$ sites which are linked together according to the Voronoi-Delaunay prescription. For each lattice size quenched averages are performed over 96 realizations. By using reweighting techniques and finite-size scaling analyses we investigate the critical properties of the model in the close vicinity of the phase transition point. Our random lattice data provide strong evidence that, for the available system sizes, the resulting effective critical exponents are indistinguishable from recent high-precision estimates obtained in Monte Carlo studies of the Ising model and ${\ensuremath{\varphi}}^{4}$ field theory on three-dimensional regular cubic lattices.We report single-cluster Monte Carlo simulations of the Ising model on three-dimensional Poissonian random lattices with up to $128000\ensuremath{\approx}{50}^{3}$ sites which are linked together according to the Voronoi-Delaunay prescription. For each lattice size quenched averages are performed over 96 realizations. By using reweighting techniques and finite-size scaling analyses we investigate the critical properties of the model in the close vicinity of the phase transition point. Our random lattice data provide strong evidence that, for the available system sizes, the resulting effective critical exponents are indistinguishable from recent high-precision estimates obtained in Monte Carlo studies of the Ising model and ${\ensuremath{\varphi}}^{4}$ field theory on three-dimensional regular cubic lattices.

arXiv:cond-mat/0211446Physical review. B, Condensed matterOct 31, 200231 citations

1 High precision single-cluster Monte Carlo measurement of the critical exponents of the classical 3D Heisenberg model ∗

Christian Holm, Wolfhard Janke E, Höchstleistungsrechenzentrum Forschungszentrum Jülich

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.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.

Jan 1, 19925 citations

Static critical behavior of three-dimensional classical Heisenberg models: A high-resolution Monte Carlo study

Kun Chen, Alan M. Ferrenberg, David Paul Landau

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).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).

Physical review. B, Condensed matterAug 1, 1993331 citations

{100}^{3}

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