Citation: Cardy, John L. "Conformal invariance and the Yang-Lee edge singularity in two dimensions." Physical review letters 54.13 (1985): 1354.
Web: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.54.1354
Tags: Physical, Conformal-field-theory, Fibonacci-anyons
In this lovely paper, John Cardy takes a look at the conformal structure of the critical edge theory of the Yang-Lee model. The story of what he finds is fascinating, and I have a lot to say about it. Here's how it goes, roughly. In earlier foundational works, Yang and Lee observed that the zeroes of the grand partition function on an analytically-continued complex plane of a phase diagram can often be used to understanding the analytic behavior of the phase diagram on the real axis. For instance, when considering the Ising model in two dimensions, understanding the zeroes of the partition function as the strength of the magnetic field varies over the complex plane helps for understanding the analytic behavior of the partition function of the real-magnetic field partition function. To quote Yang and Lee: "The problem is approached by allowing the fugacity to take on complex values. Although only real values of the fugacity are of any physical interest, the analytical behavior of the thermodynamic functions can only be completely revealed by going into the complex plane"
> Yang, Chen-Ning, and Tsung-Dao Lee. "Statistical theory of equations of state and phase transitions. I. Theory of condensation." Physical Review 87.3 (1952): 404.
> Lee, Tsung-Dao, and Chen-Ning Yang. "Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model." Physical Review 87.3 (1952): 410.
So, one is naturally led to studying the zeroes of the partition function of the Ising model with magnetic strength varying over the complex plane. Yang and Lee showed that these zeros are concentrated around the imaginary axis. More specifically, in this case, the partition function of the theory with real magnetic field can be written as a weighted integral of density of zeros on the imaginary axis (i.e the number of zeros per unit interval in the large-lattice limit). The central problem thus becomes to compute this density of zeros. In the paramagnetic phase of the Ising model, this density of zeros has a gap around Im(z)=0, becomes non-zero at some critical value, and then becomes increasingly small via some power law decay. The central problem now becomes to compute the exponential for this power law decay. Numerically this decay was seen to have exponent —0.12+0.05, and seemed to be independent of the specific choice of lattice (square lattice, triangular lattice, diamond lattice):
> Kortman, Peter J., and Robert B. Griffiths. "Density of zeros on the Lee-Yang circle for two Ising ferromagnets." Physical Review Letters 27.21 (1971): 1439.
It was then argued by Michael Fisher that these scaling constants really are universal, and are related to the scaling laws in a certain quantum conformal theory describing this critical imaginary magnetic field where the density-of-zeros becomes non-zero. This allows Fisher to make several non-trivial claims about the theory, and allowed him to make a refined numerical computation of the critical exponent (which was now shown to be universal), -0.155+0.010, which are related to the conformal invariants of the theory (central charge).
> Fisher, Michael E. "Yang-Lee edge singularity and ϕ 3 field theory." Physical Review Letters 40.25 (1978): 1610.
In the present paper, John Cardy shows how to compute sigma exactly: it is -1/6, consistent with all of the previous numerics. What's amazing is not just the computation, but also the proof technique. What Cardy does first is to consider all of the primary fields in the theory. He argues by very general principles that the primary fields of this CFT are generated by two primary fields one of which is the trivial field, and one of which is non-trivial and he calls Phi. He then argues by some abstract renormalization-group argument that the field Phi^2 is dependent on 1 and Phi, and so Phi^2 = 1 + Phi. From finding the fusion rules, Cardy can then use the algebraic rigidity of the fusion rings to make non-trivial results. That is, "The only minimal model with such simple behavior is M(5, 2)". He can thus identify exactly the relevant CFT, and from that deduce the scaling exponent is -1/6.
This is significant for several reasons. For one, this is the birth of the algebraic understating of this critical CFT. Since it takes place in the complex plane, it is necessarily a non-unitary CFT, and corresponds to a non-unitary modular category: the Yang-Lee category. This is the birth of the Yang-Lee category, though still quite far from category theory.
Another significance of this story is that it highlights the power of classification results in physics. Sometimes you can only extract a few facts about your data, but if that data has to fit into a restrictive enough algebraic structure then you can identify exactly which structure it corresponds to.
A very good discussion of this stuff is found in section 7.4.1 of
> Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal field theory. Springer Science & Business Media, 2012.