The momentum-shell method

The method discussed in Section 11.3.1 will now be pursued further, in that it will be applied to the full effective Hamiltonian of an O(n) spin system, in which the effective Hamiltonian contains terms that are linear, quadratic, and of fourth order in the spin field. It is in the consideration of the higher order terms in the effective Hamiltonian (higher order than quadratic, that is) that the complications arise. The calculations that will be outlined here are not especially challenging in execution, but we will hint at extensions and generalizations that can become so.

In this chapter, the reader will be introduced to the field-theoretical version of the renormalization group, and to its first effective realization, the ∈ expansion for critical exponents. The approach will be that of the momentum-shell method developed in the previous chapters. The application of the method to the full O(n) Hamiltonian will be more complicated due to the coupling terms, which were neglected before.A straightforward, though somewhat tedious, calculation will lead to a modified set of renormalization equations. These differential equations will then be solved to lowest order in the variable ∈ = 4 – d, where d is the system's spatial dimensionality (three in the cases of interest to us). Using scaling arguments, the critical exponents will be obtained. Their relevance to the self-avoiding random walk will also be discussed.

The techniques to be discussed here are descendants of the original renormalization group method developed by Kenneth Wilson (Wilson, 1971a; Wilson, 1971b; Wilson and Kogut, 1974).