Oxford University Press eBooks

Ferromagnetism: a renormalization group tutorial

Tom Lancaster, Stephen J. Blundell

Published Apr 17, 2014• 0 citations

Abstract

This chapter treats the ferromagnetic transition as a further example of the renormalization group procedure and shows how critical exponents can be extracted.

DOI: 10.1093/acprof:oso/9780199699322.003.0036

Related papers

The renormalization group

Tom Lancaster, Stephen J. Blundell

The renormalization group is introduced in this chapter. Examples treated include asymptotic freedom, Anderson localization, and the Kosterlitz–Thouless transition.

Oxford University Press eBooksApr 17, 20140 citations

Renormalization group approach to statistical systems

Patrik Lidén

This report will study the theory of renormalization group used on a ferromagnet to calculate magnetic phase transitions. As the ferromagnet's temperature changes, so does interatomic interaction, causing either a lingering magnetization at low temperatures, or a random state at high temperatures. At the phase transition, where the magnetization first appears/disappears, depending from which side the critical temperature is approached, the ferromagnet will behave in a certain way that causes some quantities to receive power law behaviours, so called critical exponents. The theory of renormalization group is used to find the critical point and these power units on two different two-dimensional lattices, the square lattice and the triangular lattice. The first step is to study how atoms couple to each other in one-dimension. It is possible to group atoms together into blocks. These blocks will interact with each other through new couplings. It is possible to calculate the critical point and critical exponents by comparing the block couplings with the former atomic couplings and then find the exponents. The theory of renormalization does seem to work in general, as the result will be able to explain experimentally observed relation between exponents.

Jan 1, 20150 citations

Implementations of the renormalization group

Julia M. Yeomans

Abstract In Chapter 8 we gave an introduction to the formalism of the renormalization group. A renormalization transformation changes the scale of a system by a factor b. Systems at criticality flow to fixed points under the transformation and their critical exponents and scaling behaviour are determined by linearizing the recursion equations about the fixed point. The aim of this chapter is two-fold: firstly to illustrate these ideas by giving several examples of renormalization group transformations and secondly to demonstrate the ways in which they can be used to calculate critical properties. Our examples are divided into three classes. In Section 9.1, we shall consider a one-dimensional model where the recursion relations can be written down exactly. The drawback of this as a canonical example is that any critical behaviour is at zero temperature. The decimation transformation presented in one dimension is an example of a real-space transformation and the second task, in Section 9.2, is to show how these may be used approximately to treat models in higher dimensions. Thirdly, in Section 9.5, we shall mention the E-expansion where the renormalization takes place in momentum space. This enables an understanding of the crossover to mean-field behaviour. However, the formalism is somewhat involved and the details are beyond the scope of this book.

May 7, 19920 citations

Renormalization Group and Critical Exponents for a Two‐Dimensional Ising Ferromagnet with Long‐Range Interaction

S. Fakioǧlu

physica status solidi (b)Jan 1, 19823 citations

Renormalization Group

Francesco Caravelli, Diego A. R. Dalvit

In the previous chapters, we introduced the scaling hypothesis and saw how Kadanoff's block-spin idea provides a conceptual bridge between microscopic models and macroscopic critical behavior. We also saw how Widom's scaling form and the resulting relations among critical exponents emerge from the assumption of scale invariance near the critical point. We are now ready to formalize these ideas into a mathematical framework: the renormalization group (RG).

CRC Press eBooksJun 19, 20260 citations

Phase transitions and renormalisation group

Jean Zinn‐Justin

1. Quantum Field Theory and Renormalization Group 2. Gaussian Expectation Values. Steepest Descent Method . 3. Universality and Continuum Limit 4. Classical Statistical Physics: One Dimension 5. Continuum Limit and Path Integral 6. Ferromagnetic Systems. Correlations 7. Phase transitions: Generalities and Examples 8. Quasi-Gaussian Approximation: Universality, Critical Dimension 9. Renormalization Group: General Formulation 10. Perturbative Renormalization Group: Explicit Calculations 11. Renormalization group: N-component fields 12. Statistical Field Theory: Perturbative Expansion 13. The sigma4 Field Theory near Dimension 4 14. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15. The Non-Linear sigma-Model 16. Functional Renormalization Group Appendix 1. Quantum Field Theory and Renormalization Group 2. Gaussian Expectation Values. Steepest Descent Method . 3. Universality and Continuum Limit 4. Classical Statistical Physics: One Dimension 5. Continuum Limit and Path Integral 6. Ferromagnetic Systems. Correlations 7. Phase transitions: Generalities and Examples 8. Quasi-Gaussian Approximation: Universality, Critical Dimension 9. Renormalization Group: General Formulation 10. Perturbative Renormalization Group: Explicit Calculations 11. Renormalization group: N-component fields 12. Statistical Field Theory: Perturbative Expansion 13. The sigma4 Field Theory near Dimension 4 14. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15. The Non-Linear sigma-Model 16. Functional Renormalization Group Appendix

Oxford University Press eBooksJan 1, 20139 citations