9.1: The Program of Renormalization

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A statistical mechanical system is defined by a set of degrees of freedom and by a set of coupling constants $\{K\ns_\alpha\}$. The degrees of freedom can be discrete, such as Ising spins $\sigma\ns_i$, or continuous, such as a field $\phi(\Br)$. Additionally, each such system possesses a microscopic length scale $\ell$. For discrete, lattice-based systems, this length scale is simply the lattice spacing: $\ell=a$. For continuous systems, we can define a microscopic length scale by imposing a cutoff $\RLambda$ on the wavevectors we integrate over in all Fourier transforms. That is, we replace $$\int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk) \longrightarrow \int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk)\,\gla(\Bk)\ ,$$ where $F(\Bk)$ is any function and $\gla(\Bk)$ is the cutoff function. The simplest such case to imagine is a sharp cutoff which is isotropic in wavevector, $\gla(\Bk)=\RTheta(\RLambda-|\Bk|)$. Other cutoff schemes, however, are possible, including ‘soft cutoffs’ where $\gla(\Bk)$ is smooth. The microscopic length scale is then $\ell\sim\RLambda^{-1}$, which is the smallest distance in real space over which the system can independently fluctuate.

The idea behind renormalization is that we can successively winnow degrees of freedom from a system in some exact or approximate way, and in so doing we generate a new version of the system, at a different length scale $\ell'>\ell$, and with different couplings $\{K'_\alpha\}$. We then iterate this procedure. The result is a set of equations which tell us how the couplings behave under a change of the microscopic length scale. As we shall see, the fixed points of this procedure – where couplings do not change under a change of length scale – are critical points. Such a fixed point is defined by a set of couplings $\{K^*_\alpha\}$.

If we denote by $\CR\ns_b$ the renormalization procedure $$\CR\ns_b\big(\ell\,,\{K\ns_\alpha\}\big) = \big(\ell'\,,\{K'_\alpha\}\big)\ ,$$ where $\ell'=b\,\ell$, then we have the composition law $\CR\ns_b\,\CR\ns_{b'}=\CR\ns_{b+b'}$. The set of transformations $\{\CR\ns_b\}$ is collectively referred to as the renormalization group (RG) because of this mathematical structure. It is somewhat of a misnomer, however, since the transformations are only defined for $b\ge 1$, which means that there is no inverse operation, and hence no true group structure¹. Nevertheless we shall use the RG terminology because it has become universally accepted in the literature.