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arXiv:2601.12018 · Computer Physics Communications

Padé approximation and partition function zeros

R. G. M. Rodrigues

Published May 20, 2026• 0 citations
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Abstract

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.

DOI
10.1016/j.cpc.2026.110234

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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.
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.
Zeros of the partition function were introduced by Fisher as an alternative way to study phase transitions. However, this approach has arisen some issues, the analytical treatment is diffcult and often impossible, requiring numerical methods. Despite the technologicaldevelopment, we still face two major problems for the Fisher zeros numerical solution, the polynomial generated by the canonical partition function has the coeffcient given by density of states and their degree depends on the size of the system. Then we have to deal with high degree polynomials at the order of 30.000 and the imprecision generated by numerical computation using the density of states, which has the difference between their maximal and minimal value at the order of 10400. Aiming the solution to those problems, an alternative method has been proposed [1] based on the Fisher zeros and able to solve these issues. After a simple transformation , the polynomials coeffcients are given by the energy of the system, which has values between [0, 1], if correctly normalized. This way, we have a clear criterion to select the most relevant coeffcients of the polynomial, drastically reducing their degree. In this work, we apply this new method to Ising model and two-dimensional XY, analyzing the effects of the chosen cut in the histogram anddiscussing the behaviour of the proposal algorithm in [1].
We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multipoint Padé approximants. We perform simulations for the 2D Ising model with ferromagnetic couplings both in the absence and in the presence of a magnetic field using a cluster spin-flip algorithm. We show that it is possible to extract genuine signature of Lee-Yang and Fisher zeros of the theory through the poles of magnetization and specific heat, using the multipoint Padé method. We extract the poles of magnetization using Padé approximants and compare their scaling with known results. We verify the circle theorem associated to the well known behavior of Lee-Yang zeros. We present our finite volume scaling analysis of the zeros done at T=Tc for a few lattice sizes, extracting to a good precision the (combination of) critical exponents βδ . The computation at the critical temperature is performed after the latter has been determined via the study of Fisher zeros, thus extracting both βc and the critical exponent ν . Results already exist for extracting the critical exponents for the Ising model in two and three dimensions making use of Fisher and Lee-Yang zeros. In this work, multipoint Padé is shown to be competitive with this respect and thus a powerful tool to study phase transitions. Published by the American Physical Society 2024
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.
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.
We describe an iterative method to study the critical behavior of a system based on the partial knowledge of the complex Fisher zeros set of the partition function. The method is general with advantages over most conventional techniques since it does not need to identify any order parameter a priori. The critical temperature and exponents can be obtained with great precision. To test the method and to show how it works we applied it to the 2 D Ising and 6– states Potts models. The strategy can easily be adapted to any model, classical or quantum, once we can build the corresponding energy probability distribution.