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arXiv:2508.20084 · Journal of Statistical Mechanics Theory and Experiment

Phase transition properties via partition function zeros: the Blume–Capel ferromagnet revisited

Leïla Moueddene, Nikolaos G. Fytas, Bertrand Berche

Published Oct 1, 2025• 2 citations
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Abstract

Abstract Since the landmark work of Lee and Yang, locating the zeros of the partition function in the complex magnetic-field plane has become a powerful method for studying phase transitions. Fisher later extended this approach to complex temperatures, and subsequent generalizations introduced other control parameters, such as the crystal field. In previous works (Moueddene et al 2024 J. Stat. Mech. 023206; 2024 Phys. Rev. E 110 064144) we applied this framework to the two- and three-dimensional Blume–Capel model–a system with a rich phase structure where a second-order critical line meets a first-order line at a tricritical point. We showed that the scaling of Lee-Yang, Fisher, and crystal-field zeros yields accurate critical exponents even for modest lattice sizes. In the present study, we extend this analysis and demonstrate that simulations need not be performed exactly at the nominal transition point to obtain reliable exponent estimates. Strikingly, small system sizes are sufficient, which not only improves methodological efficiency but also advances the broader goal of reducing the carbon footprint of large-scale computational studies.

DOI
10.1088/1742-5468/ae09a4

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Abstract Since the landmark work of Lee and Yang, locating the zeros of the partition function in the complex magnetic-field plane has become a powerful method for studying phase transitions. Fisher later extended this approach to complex temperatures, and subsequent generalizations introduced other control parameters, such as the crystal field. In previous works (Moueddene et al 2024 J. Stat. Mech. 023206; 2024 Phys. Rev. E 110 064144) we applied this framework to the two- and three-dimensional Blume–Capel model–a system with a rich phase structure where a second-order critical line meets a first-order line at a tricritical point. We showed that the scaling of Lee-Yang, Fisher, and crystal-field zeros yields accurate critical exponents even for modest lattice sizes. In the present study, we extend this analysis and demonstrate that simulations need not be performed exactly at the nominal transition point to obtain reliable exponent estimates. Strikingly, small system sizes are sufficient, which not only improves methodological efficiency but also advances the broader goal of reducing the carbon footprint of large-scale computational studies.
We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.
In this paper, we study finite-size effects in the Blume–Capel model through the analysis of the zeros of the partition function. We consider a complete graph and use the behavior of the partition function zeros to elucidate the crossover from effective to asymptotic properties. While in the thermodynamic limit the exact solution yields the asymptotic mean-field behavior, for finite size systems an effective critical behavior is observed. We show that even for large systems the criticality is not asymptotic. We also present insights into how partition function zeros in different complex fields (temperature, magnetic field, crystal field) give different precision and provide us with different parts of the larger picture. This includes the differences between criticality and tricriticality as seen through the lens of Fisher, Lee–Yang, and crystal field zeros.
Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
Zeros of the partition function of the one-dimensional ferromagnetic and antiferromagnetic Blume-Capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains. The equation for the distribution of zeros of the partition function in the thermodynamic limit is derived. The distribution of the Yang-Lee and Fisher zeros are studied for a variety of values of the parameters of the model. Densities of the Yang-Lee and Fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown. The edge singularity exponents are calculated analytically for the densities of the Yang-Lee and Fisher zeros. It is found that for both cases edge singularity exponents are universal and equal to sigma = -1/2 .
The critical temperature and the thermal scaling exponent of the square-lattice Blume-Capel model are obtained as functions of fugacity by calculating the partition function zeros in the complex temperature plane of the square-lattice Blume-Capel antiferromagnet for the first time. The thermal scaling exponent y t determines the specific-heat critical exponent and the correlation-length critical exponent. The value of y t = 2 indicates a strong first-order phase transition in two dimensions whereas y t = 1 corresponds to an Ising-class second-order phase transition. The full spectrum (from y t = 2 to y t = 1) of the thermal scaling exponent for the square-lattice Blume-Capel model is evaluated for the first time. The phase diagram of the square-lattice Blume-Capel model, including the tricritical point, is also obtained, and it is compared with other results in the literature.