Partition function zeros for the Blume-Capel model on a complete graph

TL;DR

This paper analyzes the zeros of the partition function in the finite-size Blume-Capel model on a complete graph to understand the crossover from effective to asymptotic critical behavior, revealing finite-size effects and differences in zeros across complex fields.

Contribution

It provides a detailed analysis of partition function zeros in the finite-size Blume-Capel model, elucidating the crossover from effective to asymptotic critical behavior and differences across complex fields.

Findings

Finite-size effects influence critical behavior in the model.

Partition function zeros differ across complex fields, affecting precision.

Criticality remains non-asymptotic even for large systems.

Abstract

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 make use of the behaviour 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 behaviour, for finite system sizes an effective critical behaviour 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.