Phase Transitions and Renormalization Group

Paperback | March 1, 2013

byJean Zinn-Justin

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This work tries to provide an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that isencountered in second order phase transitions, near the critical temperature. In this context, we will emphasize the role of gaussian distributions and their relations with the mean field approximation and Landau's theory of critical phenomena. We will show that quasi-gaussian or mean-fieldapproximations cannot describe correctly phase transitions in three space dimensions. We will assign this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories.In this framework, the renormalization group is directly related to the renormalization process, that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. We thus discuss the renormalization group in the context of various relevant field theories. Thisleads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, we construct a general functional renormalization group, which can be used when perturbative methods are inadequate.

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This work tries to provide an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that isencountered in second order phase transitions, near t...

Professor Jean Zinn-Justin is Head of Department, Dapnia, CEA/Saclay, France.

other books by Jean Zinn-Justin

Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics

Paperback|Sep 1 2010

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Format:PaperbackDimensions:464 pagesPublished:March 1, 2013Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199665168

ISBN - 13:9780199665167

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Table of Contents

1. Quantum Field Theory and Renormalization Group2. Gaussian Expectation Values. Steepest Descent Method .3. Universality and Continuum Limit4. Classical Statistical Physics: One Dimension5. Continuum Limit and Path Integral6. Ferromagnetic Systems. Correlations7. Phase transitions: Generalities and Examples8. Quasi-Gaussian Approximation: Universality, Critical Dimension9. Renormalization Group: General Formulation10. Perturbative Renormalization Group: Explicit Calculations11. Renormalization group: N-component fields12. Statistical Field Theory: Perturbative Expansion13. The sigma4 Field Theory near Dimension 414. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit15. The Non-Linear sigma-Model16. Functional Renormalization GroupAppendix1. Quantum Field Theory and Renormalization Group2. Gaussian Expectation Values. Steepest Descent Method .3. Universality and Continuum Limit4. Classical Statistical Physics: One Dimension5. Continuum Limit and Path Integral6. Ferromagnetic Systems. Correlations7. Phase transitions: Generalities and Examples8. Quasi-Gaussian Approximation: Universality, Critical Dimension9. Renormalization Group: General Formulation10. Perturbative Renormalization Group: Explicit Calculations11. Renormalization group: N-component fields12. Statistical Field Theory: Perturbative Expansion13. The sigma4 Field Theory near Dimension 414. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit15. The Non-Linear sigma-Model16. Functional Renormalization GroupAppendix

Editorial Reviews

"The clear exposition of the main ideas and the simple and agile notation the author uses help facilitate the comprehension of the different concepts presented. Researchers familiar with statistical physics methods will find a self-contained framework to grasp the essence of quantum fieldtheory and the renormalization group and to elucidate the prominent role they play at present in physics. For this reason, this book is highly recommendable due to the insight it gives into quantum field theories, providing sound basis for further research." --Journal of Statistical Physics