Padé Approximation and Partition Function Zeros

TL;DR

The paper addresses the computational bottleneck of locating partition-function zeros for phase transitions, especially the XY model, by introducing a Padé approximation across Fisher, EPD, and MGF zeros. The method significantly reduces the polynomial degree and the number of zeros needed (notably for Fisher and MGF) while preserving accurate estimates of the critical temperature, including a shifted Padé variant to accelerate convergence near the dominant zeros. Validation on the 2D Ising model confirms substantial runtime savings and correct $T_c$, and application to the XY model demonstrates robustness with Fisher zeros, despite fundamental convergence challenges in EPD/MGF for this case. The results establish a practical, cost-effective framework for analyzing phase transitions via partition-function zeros, especially in systems with nontrivial cusp structures, and provide guidance on method suitability and limitations.

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 Padé approximation to systematically reduces 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, their combination 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 reductions in polynomial degree and computation time while preserving accurate estimates of the critical temperature.

Figures (10)

• Figure 1: a) Fisher zeros from the Padé approximation with $m = [20, 40, 150]$ and $n = 20$, showing convergence toward the full zero distribution as $m$ increases. b) For $m = 150$, the dominant zero is accurately captured, while overlapping roots of $P_{150}(x)$ and $Q_{20}(x)$ are excluded as invalid points.
• Figure 2: Fisher zeros from the shifted Padé approximation with $m = 30$, $n = 20$ and $z_0=0.17$. With only $30$ terms in the approximation it was possible to identify the dominant zero.
• Figure 3: a) Distribution of EPD zeros from the Padé approximation with $m = 60$ and $n = 2$. In this case, the required polynomial order is nearly equal to the original EPD method, and a cluster of $Q_2$ zeros appears near the dominant zero, showing the necessity of a higher $m$ value. b) Only for $m = 130$ the dominant zero is correctly estimated.
• Figure 4: Padé approximation applied to the MGF method using $m=60$ and $n=60$, corresponding to half the polynomial degree of the original formulation.
• Figure 5: Two samples of Fisher zeros for the XY model with lattice size $L=50$. The phase transition is identified from the cusp structure formed by the zeros in the complex plane. Notably, the cusp is not always sharply defined.