Phase transition properties via partition function zeros: The Blume-Capel ferromagnet revisited

TL;DR

This work demonstrates that partition-function zeros (Lee–Yang, Fisher, and crystal-field) faithfully capture critical and tricritical behavior in the 2D Blume–Capel model, with reliable finite-size scaling extracted from surprisingly small lattices and even simulations away from the nominal transition. The authors validate Ising universality along the critical line by reproducing $y_t^{\rm IM}=1$ and $y_h^{\rm IM}=15/8$ across multiple zero types, and they extend the analysis to tricriticality, obtaining precise estimates for $\Delta_t$ and $y_t^{\rm TP}$ from crystal-field and Lee–Yang zeros, complemented by cumulant-based zero extraction. Density-of-zeros analyses corroborate scaling predictions $d/y_t$ and $d/y_h$ at critical and tricritical points, including crossover behavior near TP. Overall, the cumulant-zero framework enables accurate critical-program-exponent extraction from modest systems, offering substantial methodological efficiency and a scalable approach for exploring multicritical phenomena.

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, J. Stat. Mech. (2024) 023206; Phys. Rev. E 110, 064144 (2024)] 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.

Figures (18)

• Figure 1: Phase diagram of the 2D Blume-Capel model as reported in the literature. Several sets of results are shown from silvamalakiskwakZierenberg, as adapted from Zierenberg.
• Figure 2: Left panel: Simulation points in the phase diagram. Right panel: Typical spin configurations at $\Delta=0$ (spins $+1$ in red, spins $-1$ in blue, and spins $0$ in white). Top: above $T_{\rm c}$; middle: at the Ising-model transition $T_{\rm c}$; bottom: below $T_{\rm c}$.
• Figure 3: Typical spin configurations near the tricritical point (spins $+1$ in red, spins $-1$ in blue, and spins $0$ in white). Top: above $T_{\rm t}$; middle: at tricriticality; bottom: below $T_{\rm t}$. At tricriticality ($T = T_{\rm t}$), spin clusters appear fragmented and interpenetrating.
• Figure 4: Typical spin configurations near the first-order transition (spins $+1$ in red, spins $-1$ in blue, and spins $0$ in white). At a temperature below $T_{\rm t}$: top, $\Delta$ above the transition; center, at the transition; bottom, in the ordered phase. Note that for a finite system the transition is rounded, and one of the ordered configurations may appear dominated by spins that do not contribute to the magnetization, resembling a disordered phase.
• Figure 5: Finite-size scaling behaviour of the first zeros along the second-order line of the 2D Blume-Capel model. In the first two panels, the expected Ising-model renormalisation-group exponent is $y_t^{\rm IM}=1$, while in the third panel, $y_h^{\rm IM}=1.875$. CM denotes the cumulant method used to extract the zeros, which is discussed in detail below.