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

Abstract

Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method. Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size. We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality. We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.

Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

“…We now describe the cumulant method that we recently developed to determine the partition function zeros by measuring fluctuations of thermodynamic observables in systems of finite sizes [9,[32][33][34]44,45]. Here, we consider the magnetic field zeros, but the method can also be applied to other partition function zeros, for instance, the zeros in the complex plane of the inverse temperature, also known as Fisher zeros.…”

“…In addition to these theoretical developments, partition function zeros have been determined in several recent experiments [35][36][37][38][39] and additional proposals for their detection have been developed [40][41][42][43]. In the approach that we follow here, the partition function zeros are extracted from the fluctuations of the thermodynamic observable that couples to the control parameter, for instance, magnetization and magnetic field, or energy and inverse temperature [9,32,44,45]. The scheme is attractive from an experimental point of view since it makes it possible to explore phase transitions by measuring fluctuations in small systems, even for systems that are away from criticality [9,44,45].…”

“…In the approach that we follow here, the partition function zeros are extracted from the fluctuations of the thermodynamic observable that couples to the control parameter, for instance, magnetization and magnetic field, or energy and inverse temperature [9,32,44,45]. The scheme is attractive from an experimental point of view since it makes it possible to explore phase transitions by measuring fluctuations in small systems, even for systems that are away from criticality [9,44,45]. The method was used in the experiment of Ref.…”

“…2469-9950/2020/102(17)/174418 (12) 174418-1 ©2020 American Physical Society general properties of partition functions, including an important connection between the zeros of the partition function and its logarithmic derivatives, which deliver the cumulants of interest [9,32,44,45]. The idea is illustrated in Fig.…”

“…The good agreement suggests that a deep connection between Lee-Yang theory and large-deviation statistics may exist. It should be noted that we have considered some of these ideas in recent works on a simple model of a molecular zipper [9] and the Ising model in a mean-field approximation [45]. However, to further explore and strengthen these ideas we here consider the Ising model in one, two, and three dimensions as a paradigmatic example of a system that exhibits a phase transition.…”