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Physical review. B, Condensed matter

Reweighting histograms in quantum Monte Carlo studies of the Heisenberg model

Adauto J. F. de Souza, M. L. Lyra

Published Mar 1, 2002• 12 citations
Publisher

Abstract

In this communication, the histogram-reweighting technique is extended to quantum Monte Carlo studies of spin systems. The proposed method is applied to obtain equilibrium critical parameters of the quantum $S=1/2$ Heisenberg ferromagnet on simple cubic lattices. Our estimates are ${k}_{B}{T}_{c}/J=1.67764(2),$ ${U}^{*}=0.4504(1),$ $\ensuremath{\gamma}=1.397(20),$ $\ensuremath{\beta}=0.360(6),$ and $\ensuremath{\nu}=0.705(10)$, which are in perfect agreement with the best field-theory and high-temperature-series results. The method can readily be applied to general quantum-spin models, standing as an important tool for the study of quantum phase transitions.

DOI
10.1103/physrevb.65.100405

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Physical review. B, Condensed matter•Sep 1, 1991•786 citations
In this communication, the histogram-reweighting technique is extended to quantum Monte Carlo studies of spin systems. The proposed method is applied to obtain equilibrium critical parameters of the quantum S=1/2S=1/2S=1/2 Heisenberg ferromagnet on simple cubic lattices. Our estimates are kBTc/J=1.67764(2),{k}_{B}{T}_{c}/J=1.67764(2),kB​Tc​/J=1.67764(2), U∗=0.4504(1),{U}^{*}=0.4504(1),U∗=0.4504(1), \ensuremathγ=1.397(20),\ensuremath{\gamma}=1.397(20),\ensuremathγ=1.397(20), \ensuremathβ=0.360(6),\ensuremath{\beta}=0.360(6),\ensuremathβ=0.360(6), and \ensuremathν=0.705(10)\ensuremath{\nu}=0.705(10)\ensuremathν=0.705(10), which are in perfect agreement with the best field-theory and high-temperature-series results. The method can readily be applied to general quantum-spin models, standing as an important tool for the study of quantum phase transitions.
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).
Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.
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.
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.
We investigate the critical relaxational dynamics of the S=1/2S=1/2S=1/2 Heisenberg ferromagnet on a simple cubic lattice within the Handscomb prescription, on which a diagrammatic series expansion of the partition function is computed by means of a Monte Carlo procedure. Using a phenomenological renormalization-group analysis of graph quantities related to the spin susceptibility and order parameter, we obtain precise estimates of the critical exponents relations $\ensuremath{\gamma}/\ensuremath{\nu}=1.98\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.512\ifmmode\pm\else\textpm\fi{}0.002$ and of the Curie temperature ${k}_{B}{T}_{c}/J=1.6778\ifmmode\pm\else\textpm\fi{}0.0002.$ The critical correlation time ${\ensuremath{\tau}}_{\mathrm{int}}$ of both energy and susceptibility is also computed. We found the number of Monte Carlo steps needed to generate uncorrelated diagram configurations scales with the system's volume. We estimate the efficiency of the Handscomb method, comparing its ability in dealing with the critical slowing down to that of other quantum and classical Monte Carlo prescriptions.
Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 and 12\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is Kc{\mathit{K}}_{\mathit{c}}Kc​=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.