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Journal of Physics Conference Series

The impact of fluctuations on the zeros of the energy probability distribution

R. G. M. Rodrigues, L. A. S. Mόl

Published Feb 1, 2020• 2 citations
Publisher

Abstract

Abstract In this communication we explore the possible effects of statistical fluctuations on the use of the Energy Probability Distribution (EPD) zeros to study phase transitions. In the EPD zeros technique one has to find the roots of a polynomial whose coefficients are given by the EPD - a histogram of energy values obtained in a Monte Carlo simulation, for example. Phase transitions are signalized by the presence of a zero that approaches the point (1,0) in the complex plane. Once the EPD estimations are usually subject to statistical fluctuations and polynomial roots are known to be sensitive to modifications in its coefficients, we have compared the roots of a given polynomial with the roots of a perturbed one, searching for possible impacts on the method. Our results show that although the overall map of zeros is modified, the location of the dominant zero, the one that indicates the presence of a phase transition, is not affected. Indeed, even for 30% perturbation only small modifications in the dominant zero location is observed.

DOI
10.1088/1742-6596/1483/1/012007

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Abstract In this communication we explore the possible effects of statistical fluctuations on the use of the Energy Probability Distribution (EPD) zeros to study phase transitions. In the EPD zeros technique one has to find the roots of a polynomial whose coefficients are given by the EPD - a histogram of energy values obtained in a Monte Carlo simulation, for example. Phase transitions are signalized by the presence of a zero that approaches the point (1,0) in the complex plane. Once the EPD estimations are usually subject to statistical fluctuations and polynomial roots are known to be sensitive to modifications in its coefficients, we have compared the roots of a given polynomial with the roots of a perturbed one, searching for possible impacts on the method. Our results show that although the overall map of zeros is modified, the location of the dominant zero, the one that indicates the presence of a phase transition, is not affected. Indeed, even for 30% perturbation only small modifications in the dominant zero location is observed.
Partition function zeros play a central role in the study of phase transitions. Recently, energy probability distribution (EPD) zeros were proposed as an alternative approach that solves some of the implementation issues present in the Fisher zeros method by allowing drastic reduction of the polynomial. Here, a formulation based on the EPD zeros that can reduce even more the polynomial degree while maintaining its accuracy is presented. This method has shown to be computationally cheaper than the EPD zeros, allowing the study of systems by using partition function zeros that would be unfeasible otherwise. In addition, the method can be easily extended to study phase transitions in external fields while maintaining all of its improvements.
In this work, we present a computational procedure to locate the dominant Fisher zero of the partition function of a thermodynamic system. The procedure greatly reduces the required computer processing time to find the dominant zero when compared to other dominant zero search procedures. As a consequence, when the partition function results in very large polynomials, the accuracy of the results can be increased, since less drastic truncation of the polynomials (or even no truncation) is necessary. We apply the procedure to the 2D Ising model in a square lattice, obtaining very accurate results for the critical temperature and some of the critical exponents of the model. We also show the results obtained when the technique is used with the Monte Carlo simulated 2D Ising model in large lattices.
Fisher zeros play a central role in the theoretical understanding of phase transitions. However, their computation requires knowledge of the density of states, which limits their practical applicability. Alternative approaches based on the Energy Probability Distribution (EPD) and Moment Generating Function (MGF) alleviate the computational cost but suffer from convergence issues in the two-dimensional anisotropic Heisenberg model (XY model). In this work, we introduce a Padé approximation to systematically reduce the number of zeros required in the Fisher, EPD, and MGF formulations without loss of accuracy. Moreover, since the Fisher zeros formulation does not rely on a convergence algorithm, combining this approach with a Padé approximation enables a reliable analysis of the XY model while significantly reducing computational cost. Applications to the two-dimensional Ising and XY models demonstrate substantial decreases in polynomial degree and computation time while preserving accurate estimates of the critical temperature.
The concept of the partition function zeros is one of the most efficient methods for investigating the phase transitions and the critical phenomena in various physical systems. Estimating the partition function zeros requires information on the density of states Ω( E ) as a function of the energy E . Currently, the Wang-Landau Monte Carlo algorithm is one of the best methods for calculating Ω( E ). The partition function zeros in the complex temperature plane of the Ising model on an L × L square lattice ( L = 10 ~ 80) with a periodic boundary condition have been estimated by using the Wang-Landau Monte Carlo algorithm. The efficiency of the Wang-Landau Monte Carlo algorithm and the accuracies of the partition function zeros have been evaluated for three different, 5%, 10%, and 20%, flatness criteria for the histogram H ( E ).
This contribution is devoted to cover some technical aspects related to the use of the recently proposed energy probability distribution zeros in the study of phase transitions. This method is based on the partial knowledge of the partition function zeros and has been shown to be extremely efficient to precisely locate phase transition temperatures. It is based on an iterative method in such a way that the transition temperature can be approached at will. The iterative method will be detailed and some convergence issues that has been observed in its application to the 2D Ising model and to an artificial spin ice model will be shown, together with ways to circumvent them.
Thermal or finite-size scaling analyses of importance sampling Monte Carlo computer simulations in the vicinity of phase transition points often combine different estimates for the same quantity, such as a critical exponent, with the intent to reduce statistical fluctuations. We point out that the origin of such estimates in the same time series often results in pronounced cross correlations which are usually ignored even in high-precision studies, generically leading to significant underestimation of statistical fluctuations. We suggest to use a simple extension of the conventional analysis taking correlation effects into account, which leads to improved estimators with often substantially reduced statistical fluctuations at almost no extra cost in terms of computation time.