Scaling Theories and the Renormalization Group

Michael Ellis Fisher

Luca Di Carlo

Renormalization group theory is a powerful and intriguing technique with a wide range of applications. One of the main successes of renormalization group theory is the description of continuous phase transitions and the development of scaling theory. Most courses on phase transitions focus on scaling and critical exponents, while less attention is paid to universality, renormalization group flow, and the existence of a unique fixed point, which are the ultimate reasons why scaling theory is so effective in describing continuous phase transitions. We use a combination of Monte Carlo simulations and real space renormalization group theory to determine the renormalization group flow and to show the existence of a universal fixed point in the context of the ferromagnetic Ising model.

Eduardo Fradkin

Michael Ellis Fisher

The nature and origins of renormalization group ideas in statistical physics and condensed matter theory are recounted informally, emphasizing those features of prime importance in these areas of science in contradistinction to quantum field theory, in particular: critical exponents and scaling, relevance, irrelevance and marginality, universality, and Wilson's crucial concept of flows and fixed points in a large space of Hamiltonians. Contents Foreword 1 Introduction 2 Whence came renormalization group theory? 3 Where stands the renormalization group? 4 Exponents, anomalous dimensions, scale invariance and scale dependence 5 The challenges posed by critical phenomena 6 Exponent relations, scaling, and irrelevance 7 Relevance, crossover, and marginality 8 The task for renormalization group theory 9 Kadanoff's scaling picture 10 Wilson's quest 11 The construction of renormalization group transformations: the epsilon expansion 12 Flows, fixed points, universality and scaling 13 Conclusions Acknowledgments Selected bibliography Appendix A Asymptotic behavior Appendix B Unitarity of the renormalization group Appendix C Nature of a semigroup Foreword It was a pleasure to participate in the Colloquium cosponsored by the Departments of Philosophy and of Physics at Boston University “On the Foundations of Quantum Field Theory.” In the full title, this was preceded by the phrase: “A Historical Examination and Philosophical Reflections,” which set the aims of the meeting. Naturally, the participants were mainly high-energy physicists, experts in field theories, and interested philosophers of science.

Jean Zinn‐Justin

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].

Moorad Alexanian