Partition function zeros/Lee-Yang theory:

Lee-Yang theorem, derived in 1952, has been one of the most beautiful but also most perplexing mathematical result concerning phase transitions in lattice systems. In is basic version (later generalized in the works of Ruelle, Newman, Lieb-Sokal) it states that all zeros of the partition function of the Ising model with free or periodic boundary condition lie on the unit circle in the complex plane where the complex parameter is the exponential of an external field. Lee and Yang also recognized the significance of these zeros for the understanding of phase transitions. Indeed, they noted that non-analyticity of physical quantities will develop only when the set of the zeros pinches the physical domain of the relevant parameter space. However, despite the beauty and conceptual appeal of this theory, the problem of finding partition function zeros explicitly has proved to be so intractable that Lee-Yang program has failed to be implemented in any practical study of phase transitions. (An exception to this undisputed rule are perhaps the so called Griffiths singularities.)

A partial completion of the Lee-Yang program has been achieved in a joint work with C. Borgs, J.T. Chayes, L.J. Kleinwaks and R. Kotecký. Specifically, we have been able to describe the zeros of the torus partition function in any lattice spin model which has a convergent contour expansion (and which satisfies certain nondegeneracy conditions). This includes, in particular, the low-temperature Ising, Potts and Blume-Capel models in a complex external field. Among the main results, we give equations describing the positions of the zeros up to an error which is exponentially small in the linear size of the system and show that these asymptotically cover the lines of complex phase diagram. In systems possessing an Ising-like plus-minus symmetry we also establish a local version of the Lee-Yang theorem.

Relevant papers:

M. Biskup, C. Borgs, J.T. Chayes, L.J. Kleinwaks and R. Kotecký, General theory of Lee-Yang zeros in models with first-order phase transitions, Phys. Rev. Lett. 84 (2000) 4794-4797.  pdf

M. Biskup, C. Borgs, J.T. Chayes, L.J. Kleinwaks and R. Kotecký, Partition function zeros at first-order phase transitions: A general analysis, submitted to Commun. Math. Phys.  pdf TeX

M. Biskup, C. Borgs, J.T. Chayes, and R. Kotecký, Partition function zeros at first-order phase transitions: Pirogov-Sinai theory, submitted to J. Statist. Phys.  pdf TeX