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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

Published Jan 1, 1992• 5 citations
OpenAlex

Abstract

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.

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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.

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

Christian Holm, W Janke

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.

arXiv:hep-lat/9306020•Journal of Physics A Mathematical and General•Apr 7, 1994•38 citations
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.
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.
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).
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 study the efficiency and theory behind various Markov chain Monte Carlo update methods (later MCMC) in the classical Heisenberg model in three dimensions. Classical Heisenberg model is a model in statistical physics that describes ferromagnetic phenomena. Classical Heisenberg model is a generalization of the Ising model where the spin is three dimensional unit vector instead of scalar -1 or 1. Both models show a second order phase transition which is the main reason we are interested in these models. The transition in our case describes the loss of magnetization of a ferromagnet as it is heated to its Curie temperature. Monte Carlo simulating the Classical Heisenberg model uses the same MCMC update methods as the Ising model. We introduce the theoretical background of Metropolis, Overrelaxation, and Wolff single cluster updates and study the dynamic critical exponents of Metropolis and Wolff updates. Results of this study are as expected: metropolis suffers from critical slowing down near the phase transition temperature and the autocorrelation time scales to L^5.08 , where L^3 is the size of the lattice. Wolff single cluster update avoids critical slowing down and scales very well with autocorrelation time scaling to L^3.28 . Even though Wolff update scales much better it has its downsides. Parallelization is a very important factor in modern scientific computing and Wolff update is very tricky to parallelize where as Metropolis parallelizes very well. Usually we can avoid this problem by running multiple instances of the simulation at the same time as varying simulation parameters such as the temperature. For large lattices the problems is that getting moderate results requires a lot of time.
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.