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Embrace the Chaos

Sean Washburn

Supersymmetry in Disorder and Chaos. Konstantin Efetov. 441 pp. Cambridge University Press, 1996. $105.

Supersymmetry in Disorder and Chaos is a very long and demanding "text" on the calculation method that Konstantin Efetov has employed with great success for the past 15 years to classify, explain and predict many varieties of the electronic behavior of materials. In effect, it is a 450-page review article of more than a decade of work on topics at the prow of theoretical physics in condensed systems. Owing to the breadth and depth of the book, the learning curve is steep throughout. One who survives this gauntlet obtains a compact notation and a new technique for calculations that are arduous or impossible by other means. More importantly, one obtains a unified viewpoint from which to derive results on a variety of topics (such as the metal-insulator transition, polymer entanglements and chaotic scattering of electrons).

The book is a terse, formidable presentation by a formidable mind. Efetov breezes through the calculus of Grassman variables in 20 pages. The next 20 pages review perturbation analysis of density and current correlation functions with Green functions. Each of these topics has been expounded in whole monographs, so clearly this is not a book for the faint of heart. This brisk presentation is the norm for the entire book: The reader must have stamina as well as training. To read this book fruitfully, one needs to have mastered calculus, complex analysis and volumes 1, 2, 3, 5, 9 and 10 of the Landau and Lifshitz series of graduate physics texts. In other words, one must be an accomplished student of theoretical physics. In addition, a working knowledge of forefront topics in condensed matter physics is needed for context and background. The upshot? The book is not really a physics text, since anyone capable of reading the book will have learned the physics before. It is an excellent starting point to develop facility with the mathematical techniques.

Since the early 1970s, elementary particle theorists have used a method that includes a "supersymmetry" between bosons and fermions; it appears in the calculations as a "supervector" of both commuting variables and anticommuting variables. The two kinds of variables refer to the two kinds of particles. Although there are formal conversions between boson excitations and fermion excitations in theoretical models, the technique in the book has nothing to do with fundamental symmetry among particles. It is employed as an artifice to facilitate averaging and counting of states or trajectories.

There are a couple of ways that the book could have been improved. One learns most easily by relating the new to the familiar. Efetov makes less effort than he could in this direction. In fact he pointedly ducks the question with an analogy to imaginary numbers saying one needn't know "the physics of ??1—" to use it profitably in calculation—that one should accept the mathematical technique on faith. This is a peculiar viewpoint to express to an audience of physicists, where even the undergraduates know that ??1— is used as a bookkeeping tool to catalogue relative phases of solutions to equations. This omission of qualitative physical understanding and the misquoting of a few names (Jerome Licini and Serge Luryi, for example) appear to be the only flaws in the presentation. Ultimately the book is engaging and successful. Full of wit, sharp insight and deep physics, it follows the trend at Cambridge (which includes the recent advanced texts by Tsvelik and by Chaikin and Lubensky). Efetov's English is lively and colloquial. His opinions about needless mathematical rigor and shoddy physical reasoning boil over from time to time. My personal favorites are his reference to "complicated mathematical notations that very often frighten physicists'' and the remark in the chapter on metal-insulator transitions that a certain (Nobel-prize-winning) argument can be considered "a joke'' in light of his analysis. How can one fail to be charmed by someone willing to fire salvos like that?—Sean Washburn, Physics and Astronomy and Applied Sciences, University of North Carolina at Chapel Hill

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