Large-size critical behavior of infinitely coordinated systems

Abstract

Details are presented of an extension of the size-scaling hypothesis to systems in which each element interacts equally with all others (systems for which the mean-field approximation is valid in the thermodynamic limit). A simple argument, which relates the large-size critical behavior of physical quantities with the upper critical dimensionality of the corresponding short-range system, already presented in a Letter, is here made precise and checked either analytically or numerically on several examples. In particular, the scaling form for the magnetization is explicitly derived in the case of the infinitely coordinated Ising model, and a numerical study is presented of the infinitely coordinated $\mathrm{XY}$-Ising quantum model in a transverse field, with its extension in the presence of an imaginary longitudinal field (a model exhibiting a Yang-Lee edge singularity).