Pad\'e Approximation and Partition Function Zeros

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.