Describing actual critical behaviour from field theory: a delicate matter

H. K. Janssen, B. Schmittmann

Jean Zinn‐Justin

Gernot Münster, Harald W. Grießhammer, Dirk Lehmann

These lecture notes want to illustrate the close connection between statistical mechanics and field theory not only on the formal level, i.e. that many concepts of one area can easily be taken over to the other one, but also on the level of actual calculations. To this purpose, the last section will demonstrate that a special statistical system, the binary fluid system, can be described by field theory in its critical behaviour.

Steven Weinberg

John Cardy

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.

M Cristina Diamantini, Carlo A. Trugenberger

Conformal field theories do not only classify two dimensional (2D) classical critical behaviour, but they also govern a certain class of 2D quantum critical behaviour. In this latter case, it is the ground state wave functional of the quantum theory that is conformally invariant, rather than the classical action. We show that the superconducting-insulating (SI) quantum phase transition in 2D Josephson junction arrays (JJAs) is a (doubled) c = 1 Gaussian conformal quantum critical point. The quantum action describing this system is a doubled Maxwell–Chern–Simons model in the strong coupling limit. We also argue that the SI quantum transitions in frustrated JJAs realize the other possible universality classes of conformal quantum critical behaviour, corresponding to the unitary minimal models at central charge c = 1 − 6/ m ( m + 1) for m ⩾ 3.