On spinodal points and Lee-Yang edge singularities

We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the phi(4) theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid-gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e. T < T-c, and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field H for T > T-c. The extended analyticity conjecture (due to Fonseca and Zamolodchikov) posits that, for T < T-c, the Lee-Yang edge singularities are the closest singularities to the real H axis. This has interesting implications, in particular, that the spinodal singularities must lie off the real H axis for d < 4, in contrast to the commonly known result of the mean-field approximation. We find that the parametric representation of the Ising equation of state obtained in the epsilon = 4-d expansion, as well as the equation of state of the O(N)-symmetric phi(4) theory at large N, are both nontrivially consistent with the conjecture. We analyze the reason for the difficulty of addressing this issue using the epsilon expansion. It is related to the long-standing paradox associated with the fact that the vicinity of the Lee-Yang edge singularity is described by Fisher's phi(3) theory, which remains nonperturbative even for d -> 4, where the equation of state of the phi(4) theory is expected to approach the mean-field result. We resolve this paradox by deriving the Ginzburg criterion that determines the size of the region around the Lee-Yang edge singularity where mean-field theory no longer applies.