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

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.