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arXiv:cond-mat/0401178 · International Journal of Modern Physics C

PROPERTIES OF ITERATIVE MONTE CARLO SINGLE HISTOGRAM REWEIGHTING

MARTIN GMITRA, Denis Horváth

Published Dec 1, 2005• 1 citation
arXivHTMLPDF

Abstract

We present an iterative Monte Carlo algorithm for which the temperature variable is attracted by a critical point. The algorithm combines techniques of single histogram reweighting and linear filtering. The ferromagnetic 2D Ising model is studied numerically as an illustration. In that case, the iterations reach a stationary regime with an invariant probability distribution function of temperature which peaked near the pseudocritical temperature of the specific heat. The sequence of generated temperatures is analyzed in terms of stochastic autoregressive model. The error of histogram reweighting can be better understood within the suggested model. The presented model yields a simple relation, connecting the variance of pseudocritical temperature and the parameter of linear filtering.

DOI
10.1142/s0129183105008606

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We present an iterative Monte Carlo algorithm for which the temperature variable is attracted by a critical point. The algorithm combines techniques of single histogram reweighting and linear filtering. The ferromagnetic 2D Ising model is studied numerically as an illustration. In that case, the iterations reach a stationary regime with an invariant probability distribution function of temperature which peaked near the pseudocritical temperature of the specific heat. The sequence of generated temperatures is analyzed in terms of stochastic autoregressive model. The error of histogram reweighting can be better understood within the suggested model. The presented model yields a simple relation, connecting the variance of pseudocritical temperature and the parameter of linear filtering.
We present iterative Monte Carlo algorithm for which the temperature variable is attracted by a critical point. The algorithm combines techniques of single histogram reweighting and linear filtering. The 2d Ising model of ferromagnet is studied numerically as an illustration. In that case, the iterations uncovered stationary regime with invariant probability distribution function of temperature which is peaked nearly the pseudocritical temperature of specific heat. The sequence of generated temperatures is analyzed in terms of stochastic autoregressive model. The error of histogram reweighting can be better understood within the suggested model. The presented model yields a simple relation, connecting variance of pseudocritical temperature and parameter of linear filtering.
Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.
A method of reweighting Monte Carlo data to parameters other than the simulated ones is investigated. The method is also extended to include Monte Carlo renormalization-group simulations. Both single and multiple simulation reweighting is used. The models studied are the d=2 ferromagnetic q=3 Potts model and the d=2 antiferromagnetic Ising model. It is shown that the reweighting produces a systematic shift in the height and location of the specific-heat peak. This shift, which without proper care may lead to erroneous results, depends on the simulated and reweighted temperatures and on the number of configurations included in the reweighting. Only for very large numbers of configurations does the shift decrease as the inverse square root of the number of configurations. Within the Monte Carlo renormalization-group study, it is shown that the critical exponents for the Ising model are determined more accurately from a multiple reweighting using one simulation below the critical temperature and one above than from the individual simulations. It is shown that it is more efficient to use two simulations at a time than all of them at once when reweighting multiple simulations.
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), $\ensuremath{\gamma}=1.397(20),$ $\ensuremath{\beta}=0.360(6),$ 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.
The probability distribution of the order parameter is analyzed in order to obtain the criticality of magnetic systems. Monte Carlo simulations have been employed by using single spin flip Metropolis algorithm aided by finite-size scaling and histogram reweighting techniques. A method is proposed to obtain this probability distribution even when the transition temperature of the model is unknown. A test is performed on the two-dimensional spin-1/2 and spin-1 Ising model and the results show that the present procedure can be quite efficient and accurate to describe the criticality of the system.
Recent developments in the use of histograms (measured discrete probability distributions) now make it possible to determine critical properties with high accuracy using Monte Carlo (MC) simulations combined with finite-size scaling analyses. The advantages and limitations of the use of histograms for extracting the maximum information from MC simulation data are described. Results from extensive MC studies of the critical properties of two important magnetic systems, the d=3 ferromagnetic Ising and Heisenberg models, are presented. It is shown that the careful use of histogram techniques can provide results comparable or superior to these obtained with other numerical methods while maintaining all the advantages of standard MC techniques.