Introduction to the Functional Renormalization Group (Lecture Notes in Physics, 798)

Annotation This book, based on a graduate course given by the authors, is a pedagogic and self-contained introduction to the renormalization group with special emphasis on the functional renormalization group. The functional renormalization group is a modern formulation of the Wilsonian renormalization group in terms of formally exact functional differential equations for generating functionals. In Part I the reader is introduced to the basic concepts of the renormalization group idea, requiring only basic knowledge of equilibrium statistical mechanics. More advanced methods, such as diagrammatic perturbation theory, are introduced step by step. Part II then gives a self-contained introduction to the functional renormalization group. After a careful definition of various types of generating functionals, the renormalization group flow equations for these functionals are derived. This procedure is shown to encompass the traditional method of the mode elimination steps of the Wilsonian renormalization group procedure. Then, approximate solutions of these flow equations using expansions in powers of irreducible vertices or in powers of derivatives are given. Finally, in Part III the exact hierarchy of functional renormalization group flow equations for the irreducible vertices is used to study various aspects of non-relativistic fermions, including the so-called BCS-BEC crossover, thereby making the link to contemporary research topics The renormalization group (RG) has nowadays achieved the status of a meta-theory, which is a theory about theories. The theory of the RG consists of a set of concepts and methods which can be used to understand phenomena in many different ?elds of physics, ranging from quantum ?eld theory over classical statistical mechanics to nonequilibrium phenomena. RG methods are particularly useful to understand phenomena where ?uctuations involving many different length or time scales lead to the emergence of new collective behavior in complex many-body systems. In view of the diversity of ?elds where RG methods have been successfully applied, it is not surprising that a variety of apparently different implementations of the RG idea have been proposed. Unfortunately, this makes it somewhat dif?cult for beginners to learn this technique. For example, the ?eld-theoretical formulation of the RG idea looks at the ?rst sight rather different from the RG approach pioneered by Wilson, the latter being based on the concept of the effective action which is ite- tively calculated by successive elimination of the high-energy degrees of freedom. Moreover, the Wilsonian RG idea has been implemented in many different ways, depending on the particular problem at hand, and there seems to be no canonical way of setting up the RG procedure for a given problem. Front Matter....Pages i-xiv Front Matter....Pages 1-3 Phase Transitions and the Scaling Hypothesis....Pages 5-22 Mean-Field Theory and the Gaussian Approximation....Pages 23-52 Wilsonian Renormalization Group....Pages 53-89 Critical Behavior of the Ising Model Close to Four Dimensions....Pages 91-121 Field-Theoretical Renormalization Group....Pages 123-139 Front Matter....Pages 141-145 Functional Methods....Pages 147-180 Exact FRG Flow Equations....Pages 181-208 Vertex Expansion....Pages 209-232 Derivative Expansion....Pages 233-247 Front Matter....Pages 249-254 Fermionic Functional Renormalization Group....Pages 255-303 Normal Fermions: Partial Bosonization in the Forward Scattering Channel....Pages 305-326 Superfluid Fermions: Partial Bosonization in the Particle–Particle Channel....Pages 327-368 Back Matter....Pages 369-375 Part I. Foundations Of The Renormalization Group. 1. Phase Transitions And The Scaling Hypothesis ; 2. Mean-field Theory And The Gaussian Approximation ; 3. Wilsonian Renormalization Group ; 4. Critical Behavior Of The Ising Model Close To Four Dimenions ; 5. Field-theoretical Renormalization Group -- Part Ii. Introduction To The Functional Renormalization Group. 6. Functional Methods ; 7. Exact Frg Flow Equations ; 8. Vertex Expansion ; 9. Derivative Expansion -- Part Iii. Functional Renormalization Group Approach To Fermions. 10. Fermionic Functional Renormalization Group ; 11. Normal Fermions : Partial Bozonization In Forward Scattering Channel ; 12. Superfluid Fermions : Partial Bozonization In The Particle-particle Channel. P. Kopietz, L. Bartosch, F. Schütz. Includes Bibliographical References And Index.