Critical and tricritical behavior of the $d=3$ Blume-Capel model: Results from small-scale Monte Carlo simulations

TL;DR

This study uses small-scale Monte Carlo simulations to accurately locate the critical and tricritical points of the 3D Blume-Capel model by analyzing partition function zeros and magnetization cumulants, confirming theoretical critical exponents.

Contribution

It demonstrates that small system sizes in Monte Carlo simulations can effectively determine critical and tricritical points of the 3D Blume-Capel model using zero analysis and cumulants.

Findings

Accurate determination of the tricritical point using partition function zeros.

Confirmation of theoretical logarithmic corrections at the tricritical point.

Precise estimation of the critical temperature consistent with 3D Ising universality.

Abstract

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.