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Phase transitions are typically accompanied by non-analytic behaviors of thefree energy, which can be explained by considering the zeros of the partitionfunction in the complex plane of the control parameter and their approach tothe critical value on the real-axis as the system size is increased. Recentexperiments have shown that partition function zeros are not just a theoreticalconcept. They can also be determined experimentally by measuring fluctuationsof thermodynamic observables in systems of finite size. Motivated by thisprogress, we investigate here the partition function zeros for the Curie-Weissmodel of spontaneous magnetization using our recently established cumulantmethod. Specifically, we extract the leading Fisher and Lee-Yang zeros of theCurie-Weiss model from the fluctuations of the energy and the magnetization insystems of finite size. We develop a finite-size scaling analysis of thepartition function zeros, which is valid for mean-field models, and whichallows us to extract both the critical values of the control parameters and thecritical exponents, even for small systems that are away from criticality. Wealso show that the Lee-Yang zeros carry important information about the raremagnetic fluctuations as they allow us to predict many essential features ofthe large-deviation statistics of the magnetization. This finding mayconstitute a profound connection between Lee-Yang theory and large-deviationstatistics.

Lee-Yang theory of the Curie-Weiss model and its rare fluctuations