Book contents
- Frontmatter
- Contents
- Preface
- 1 A few well-known basic results
- 2 Introduction: order parameters, broken symmetries
- 3 Examples of physical situations modelled by the Ising model
- 4 A few results for the Ising model
- 5 High-temperature and low-temperature expansions
- 6 Some geometric problems related to phase transitions
- 7 Phenomenological description of critical behaviour
- 8 Mean field theory
- 9 Beyond the mean field theory
- 10 Introduction to the renormalization group
- 11 Renormalization group for the φ4 theory
- 12 Renormalized theory
- 13 Goldstone modes
- 14 Large n
- Index
12 - Renormalized theory
Published online by Cambridge University Press: 05 July 2014
- Frontmatter
- Contents
- Preface
- 1 A few well-known basic results
- 2 Introduction: order parameters, broken symmetries
- 3 Examples of physical situations modelled by the Ising model
- 4 A few results for the Ising model
- 5 High-temperature and low-temperature expansions
- 6 Some geometric problems related to phase transitions
- 7 Phenomenological description of critical behaviour
- 8 Mean field theory
- 9 Beyond the mean field theory
- 10 Introduction to the renormalization group
- 11 Renormalization group for the φ4 theory
- 12 Renormalized theory
- 13 Goldstone modes
- 14 Large n
- Index
Summary
The approach of Chapter 11, in which we have kept a large momentum or short-distance cut-off, is sufficient for the needs of computing the various universal indices and functions that characterize criticality. All calculations were made in the limit in which the ratio ξ/a of the correlation length ξ to the lattice spacing a is large. In the previous chapters, this was done by keeping a fixed and increasing ξ.
The renormalized theory relies on the opposite strategy: it deals directly with the scaling theory in which one takes the limit of a vanishingly small a. I have included here a chapter on the renormalized theory for several reasons:
• There is no ultraviolet (or lattice) cut-off any more and, getting rid of one dimensionful parameter considerably simplifies the integrations, especially at higher orders.
• All the critical exponents and scaling functions can be computed at Tc, i.e., in a massless theory, even when one considers temperature-dependent laws, such as those concerning the correlation length, the magnetic susceptibility or the spontaneous magnetization. Integrals in a massless theory without cut-off are by far simpler, and they are often given by mere dimensional analysis.
• However, the main reason is to understand the meaning of renormalizability from the viewpoint both of critical phenomena and of high-energy physics. This will be done in the following section.
- Type
- Chapter
- Information
- Introduction to Statistical Field Theory , pp. 128 - 137Publisher: Cambridge University PressPrint publication year: 2010