This book provides 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 is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns 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. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
This book provides 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 is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns 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. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
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
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.
This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.