The Lee-yang Theory of Phase Transitions

Magnetic materials in nature exhibit a sharp phase transition: there exists a temperature (the Curie temperature) just above which the spontaneous magnetism of a magnetic material is lost. The modeling of such discontinuities presented a major conundrum for statistical physics in the early part of the 20th century. One of the rst models of magnets proposed to qualitatively model this phenomenon envisioned a magnet as a lattice of points, with a magnetic “spin”—that could be aligned in a “+” or “−” direction—at each vertex of the lattice. The spins were modeled as interacting with their neighbors so that neighbors favored having the same spin, and these interactions induced a probability distribution over the possible con gurations, i.e. assignments of +/− spins to the sites. The interactions themselves were modeled as becoming weaker with increase in a temperature parameter β.1 In this picture, the magnetization M(β) of the material was modeled as the fraction of the sites that could be expected to be assigned +, so that a fraction of 1/2 signi ed no magnetism, and phase transitions were expected to manifest as discontinuities of the function M(β) or its successive derivatives. We shall formally describe the above model (which is known as the Ising model [Isi25], but was proposed by Lenz) in the next section. However, it should not be hard to imagine that in any reasonable version of the model, discontinuities in M(β) could not be expected to arise in nite graphs. One way around this obstacle is to consider the limit of M(β) as the size of the lattice grows to in nity (this is referred to as the “in nite volume” limit). The rst such calculation was done by Ising [Isi25], who considered the “one dimensional lattice” Z, and showed that that the in nite volume limit ofM(β) was also a real analytic function in this case—there were no phase transitions. This engendered skepticism2 about whether phase transitions could be expected at all in the Ising model, and whether it was a reasonable qualitative model of magnetism at all. This skepticism was not completely laid to rest till the celebrated tour-de-force of Onsager [Ons44], who explicitly calculated the in nite volume limit for the two dimensional lattice Z2, and showed that its second derivative was indeed discontinuous. However, another feature of magnetic materials is that their magnetization is in uenced by external magnetic elds, and that the magnetization does not typically show a phase transition with respect to the magnetic eld. It is not hard to account for an external magnetic eld in the Ising model; this is modeled as a bias parameter λwhich corresponds to the tendency of sites to align themselves to the + direction. The previously cited results were, however, restricted to the “zero eld” case, and shed little light on the behavior of the model under a magnetic eld. In 1952, Lee and Yang [YL52, LY52] related this problem to the study of the location of the zeros of a polynomial called the partition function that is canonically associated with such models. Their work showed that the Ising model indeed does not show a phase transition with respect to the external eld at any non-zero value of the eld. In doing so, they also founded what proved to be a very fruitful branch of the stability theory of polynomials, one that continues to be an active area of research in both mathematics and statistical physics. In this short note, we survey how this connection to stability theory arose in the work of Lee and Yang.