Pad\'e Approximation and Partition Function Zeros

TL;DR

This paper introduces a Padé approximation method to efficiently compute Fisher zeros and analyze phase transitions in the 2D Ising and XY models, reducing computational cost while maintaining accuracy.

Contribution

The authors develop a Padé approximation approach that reduces the number of zeros needed in Fisher, EPD, and MGF methods, enabling more efficient analysis of phase transitions.

Findings

Significant reduction in polynomial degree and computation time.

Accurate estimation of critical temperatures maintained.

Method applicable to both Ising and XY models.

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 \textbf{anisotropic Heisenberg} model (XY model). In this work, we introduce a Pad\'e 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 Pad\'e 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.