Introduction to the Replica Theory of Disordered Statistical Systems , Viktor Dotsenko Cambridge U. Press, New York, 2001. $74.95 (220 pp.). ISBN 0-521-77340-7
Understanding statistical systems with quenched (or frozen-in) disorder has proved surprisingly challenging. The “replica method,” proposed by Mark Kac, remains one of the few successful theoretical tools for analyzing these systems. The method involves “creating” multiple copies of the system, averaging over the disorder, and then taking the number of copies to zero. But this turns out to be the least bizarre part.
The proving ground for the replica method was the spin glass, a disordered magnetic system in which the local couplings vary randomly in sign (and possibly magnitude). In a landmark paper, Sam Edwards and Philip Anderson boiled the problem down to its physical essentials. But even David Sherrington and Scott Kirkpatrick’s further simplification to an infinite-range model proved surprisingly difficult to treat. What is now believed to be its correct solution—the replica symmetry breaking (RSB) approach of Giorgio Parisi—took several years to appear and another several years to be interpreted.
The idea of breaking the replica symmetry (that is, assuming that, somehow, different copies of the system should not be regarded as identical) was originally suggested by the work of Jairo de Almeida and David Thouless. But Parisi identified the correct pattern of RSB for the infinite-range model. His solution’s eventual interpretation, as a new type of broken symmetry with an exotic arrangement of numerous low-temperature phases, generated widespread excitement.
While a description of the RSB approach has appeared in several excellent reviews and books, those have been devoted to the larger subject of spin glasses, and replica theory is but one part of that story. In contrast, Viktor Dotsenko’s Introduction to the Replica Theory of Disordered Statistical Systems is entirely devoted to replica theory, from the point of view that the spin glass is but one part of that story.
Regardless of one’s opinions on this last point, a knowledge of the replica approach and its technical machinery is indispensable for anyone aspiring to work in the field of quenched disorder. Dotsenko’s book is therefore a welcome addition.
The book is divided into three parts: The first covers spin glasses (mostly infinite-range models); the second, the effects of quenched disorder on critical phenomena; and the third, applications of RSB to other systems. The book’s principal strength is that it provides a clear introduction to the mathematical machinery of RSB, accessible to an advanced graduate student interested in working in the statistical mechanics of disordered systems. For this reason alone it should be part of the library of any theorist working in these areas. At the same time, the book falls into the trap (particularly in the second and third parts) of descending too quickly into a heavily technical morass of detailed calculations that assume the reader already has a sophisticated statistical mechanics background. Dotsenko seems unsure whether his book is intended for a beginner or an expert.
Along with inconsistency in the level of sophistication, the text’s other problem is perhaps that of the RSB approach itself: It is quite good when it deals with mathematical procedures and algorithms, but obscure when it tries to interpret what those calculations mean physically. (At this point some truth-in-advertising is required: my collaborator Charles Newman and I have been vocal critics of traditional interpretations of RSB theory and of attempts to apply it to more realistic spin glass models. For this reason I had particular problems with the book’s discussion of the physical interpretation of RSB. But my misgivings perhaps are less the fault of the author than of the field, which has spawned a number of “urban legends” that are at best murky, often misleading, and sometimes simply incorrect.)
Dotsenko states in the preface, perhaps wisely, that he will avoid the contentious issue of the applicability of RSB theory to realistic spin glasses. But he’s not completely consistent about doing so. His discussions of aging experiments, for example, are interpreted solely from the RSB viewpoint. Such a presentation is fine, given the book’s mission, but someone new to the field might be misled into thinking that these interpretations are the only game in town. (The preface also asserts that the scaling approach of William McMillan, Alan Bray and Michael Moore, and Daniel Fisher and David Huse, will not be discussed. Why not throw this assertion to the wind as well, and at least acknowledge that some influential theoretical studies of aging start from this very different picture of the spin glass phase?)
I would also like to have seen a few more areas covered. Mean-field dynamics is barely discussed. Some recent applications of RSB theory—to quantum spin glasses, for example—are missing entirely. I hope such topics can be included in an expanded second edition.
However, I think the book’s biggest shortcoming is its bibliography. An introductory treatment should at least have an extensive, if not exhaustive, list of references. But the bibliography is highly selective and often very incomplete. Just two of many examples: the discussion of lower critical dimension of random field magnets neglects to mention the contribution of Michael Aizenman and Jan Wehr, and the Griffiths phase is covered without mentioning the work of Deepak Dhar and Mohit Randeria, James Sethna, and Richard Palmer. This kind of problem recurs throughout the book. Fortunately, it is a shortcoming that is easily corrected in a second edition.
The final chapter returns to the difficult problem of the low-temperature phase (assuming there is one) of real spin glasses: Is RSB relevant or not? A book review is perhaps not the place to address an important issue that is theoretically—if not rigorously—avoided by the book itself. Dotsenko closes by asserting his belief in the affirmative; for myself, I believe that “interesting times” have arrived for RSB. But this does not diminish my appreciation of the book’s value. RSB theory is a stunning achievement, even if more limited in applicability than some hope, and Dotsenko’s book will remain an important contribution.
