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Monte Carlo Study of Topological Defects in the 3D Heisenberg Model

Christian Holm, Wolfhard Janke E

Published Jan 1, 1994• 19 citations
OpenAlex

Abstract

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.

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Monte Carlo study of topological defects in the 3D Heisenberg model

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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 K c , 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(1988) who extrapolated a divergent behaviour of d(n)/dT at K c 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 alpha / nu of the specific-heat exponent with the correlation-length exponent from a finite-size scaling analysis of the energy.

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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 K c , 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(1988) who extrapolated a divergent behaviour of d(n)/dT at K c 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 alpha / nu of the specific-heat exponent with the correlation-length exponent from a finite-size scaling analysis of the energy.
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.
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.
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.
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).
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.