Ken Wilson, Renormalization Group, and Differential Geometry

2009 December 7
by Daniel Lakeland

I'm reading Ken Wilson's 1971 paper on Renormalization Group based solutions to the Ising model for my Statistical Mechanics class. The paper is a good example of an important idea in its early stages of exposition.

Wilson has thought up something important, but he hasn't yet worked out the best way to communicate it, or the best structure in which to interpret it. The result is a bit of back and forth between different viewpoints on the same thing.

The key insight that he seems to have had is that at the critical state the distribution of states p(s|K,h) should be invariant under a continuous transformation that has one parameter. Here s is a state vector, and K and h are given parameters of the Ising system, with K being the nearest neighbor coupling and h the external magnetic field. Whereas Kadanoff had the idea of literally combining the spins at several points on the lattice into a new set of spins (ie. downsampling the state vector s), Wilson imagines a continuous transformation in which length can be rescaled by any factor, especially a factor like (1+\delta) with \delta \ll 1. By considering the problem in this context, we can bring what we know about Lie Groups into play in solving for the behavior of the system.

I think where he is going is that the system is considered in terms of a function F=-\frac{k_bT \log(Z)}{V} which is the free energy density, closely related to the partition function Z and hence the probability distribution p(s). We imagine a 1 parameter group action on the coefficients, with the parameter being L the rescaling length. Unfortunately, Wilson does not make the argument explicitly nondimensional, and I think this is a mistake. To make this all make sense, we should nondimensionalize the system.

Consider the nondimensional variables v=V/L^3, f = FL^3/k_bT, k = (K-K_c)/K_c and b=h/k_bT which is nondimensional if we consider spins to be nondimensional. If we contemplate rescaling these variables by multiplying the reference length by (1+l), we wish to leave the resulting f unchanged, which induces a necessary change in k and b. We want, in other words, f(v,k,b) = (1+l)^{3} f(v(1+l)^{-3},k_l,b_l). Here the exponents are opposite to what Wilson uses, but that's because here we're rescaling the reference length L which appears in the denominator of v = V/L^3. The action of rescaling v induces a necessary renormalization of k, b, and f in order to maintain the nondimensional free energy density at a constant value. This simple perspective seems hidden in the Wilson paper, and I wonder whether it's not presented like that in the paper because I'm in a position of taking advantage of almost 40 years of development, or because I don't understand something about the basics?

I wanted to go on here, but I'm afraid I'll have to wait until I do understand this perspective more. Maybe some reader will have something useful to add?

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