Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction to techniques
- 2 Generating functions I
- 3 Generating functions II: recurrence, sites visited, and the role of dimensionality
- 4 Boundary conditions, steady state, and the electrostatic analogy
- 5 Variations on the random walk
- 6 The shape of a random walk
- 7 Path integrals and self-avoidance
- 8 Properties of the random walk: introduction to scaling
- 9 Scaling of walks and critical phenomena
- 10 Walks and the O(n) model: mean field theory and spin waves
- 11 Scaling, fractals, and renormalization
- 12 More on the renormalization group
- References
- Index
12 - More on the renormalization group
Published online by Cambridge University Press: 03 December 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction to techniques
- 2 Generating functions I
- 3 Generating functions II: recurrence, sites visited, and the role of dimensionality
- 4 Boundary conditions, steady state, and the electrostatic analogy
- 5 Variations on the random walk
- 6 The shape of a random walk
- 7 Path integrals and self-avoidance
- 8 Properties of the random walk: introduction to scaling
- 9 Scaling of walks and critical phenomena
- 10 Walks and the O(n) model: mean field theory and spin waves
- 11 Scaling, fractals, and renormalization
- 12 More on the renormalization group
- References
- Index
Summary
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).
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- Information
- Elements of the Random WalkAn introduction for Advanced Students and Researchers, pp. 285 - 322Publisher: Cambridge University PressPrint publication year: 2004