Random fields and spin classes. A field theory approach.

*(English)*Zbl 1138.82030
Cambridge: Cambridge University Press (ISBN 0-521-84783-4/hbk). xv, 213 p. (2006).

Disordered spin systems have been an active field of research for decades, both in the theoretical and mathematical physics communities, and to this day they have not ceased to be so. This book, written by two theoretical physicists whose competence on the subject cannot be doubted, treats two paradigmatic examples of quenched disordered systems: the Random Field Ising Model (RFIM) (Ch. 2 and 3) and Spin Glasses, both mean field (Ch. 4 to 7) and in finite dimension (Ch. 8 ff.). While some of the material treated (especially mean field spin glasses) is already covered by existing reviews or books, what makes the book precious is the place dedicated to far less standard and more controversial subjects, especially the field-theory analysis of the Edwards-Anderson model and of the RFIM. Analysis of these topics was until now essentially scattered throughout the existing (and rather cryptic, it should be said) literature. The Edwards-Anderson is studied here with the viewpoint that the Parisi replica symmetry breaking picture should be a good description of it, at least in high dimension. While this view is still very controversial, the Parisi picture is the only starting point which allows to formulate a microscopic field theory for spin glasses, as one of the authors puts it very clearly in the Preface to the book. The point of view and the language adopted by the authors is that of field theory. The reader will therefore find references to supersymmetry, dimensional reduction, Ward identities and the like. While the is no pretense of mathematical rigor, the mathematically oriented reader will find in this book an extremely clear exposition of ideas and techniques, and for sure a deep source of inspiration.

Reviewer: Fabio L. Toninelli (Lyon)

##### MSC:

82D30 | Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses) |