Electron Correlation in Metals Kosaku Yamada Cambridge U. Press, New York, 2004. $100.00 (245 pp.). ISBN 0-521-57232-0

Kosaku Yamada’s book covers various topics in metal physics in which electron correlations play an important role. Starting with the free-electron gas and Fermiliquid theory, the author discusses the Kondo effect and the Anderson and Hubbard models, with the goal of describing the physics of heavy fermions and superconductivity in the cuprates, ruthenates, and organics.

The positive features of *Electron Correlation in Metals* are its treatments of diagrammatic techniques for the different topics. The negatives are that the book’s discussions are limited to perturbative techniques, without any critical analysis of when that approach may be valid for those same topics.

In the first part of the book, the author introduces the essential formalism of Green functions and response functions and, in chapters 4 and 5, gives much of the algebraic details for the single-impurity Kondo and Anderson Hamiltonians. For the Kondo problem, he also offers a brief discussion of scaling theory.

In the second part, chapters 6 through 9, Yamada turns to models relevant to heavy fermions and high-*T* _{c} superconductivity. Unfortunately, this part is where the book has its biggest lacunae: Many aspects of the phenomena are not well understood and are likely beyond the reach of perturbative diagrammatic methods. Yamada, however, seems convinced that a combination of superconductivity mediated by spin fluctuations (FLEX approximation) can describe the entire phenomenology of the high-*T* _{c} cuprates.

The beginning sections of chapter 9 are rather flawed and unmotivated. The first section, on high-*T* _{c} superconductivity, starts out with “We explain the mechanism on the basis of Fermi liquid theory,” which is a surprising statement because the system, in the normal state, shows considerable experimental evidence for a breakdown of Fermiliquid theory. It is crucial for the reader to recognize that solving the problem of high-*T* _{c} superconductivity means understanding the physics of doped Mott insulators in the strong-correlation regime. The book’s singular focus of describing electrons that interact via spin fluctuations is really only valid at high doping. In the case of heavy fermions, Yamada does not mention recent developments in non-Fermi liquid behavior near quantum critical points.

Given the fact that beautiful data from angle-resolved photoemission spectroscopy experiments are available, Yamada could have used the information to illustrate the tight binding model and its parameter values in chapter 9. Later in the chapter, the author goes back and forth between theory and experiments mainly involving transport and nuclear magnetic resonance respectively. The mysteries of the normal state are explained away in terms of electron—electron scattering due to antiferromagnetic spin fluctuations within the Fermi liquid. However, it is not clear whether the author believes that the theory, at the present level, can explain the experiments completely or whether some puzzles remain.

The author observes that “because of the strong on-site repulsion, the symmetry of the pairing state is not the *s*-wave symmetry but an anisotropic symmetry, such as the *p*- and *d*-wave symmetry.” But he fails to mention the definitive experiments—of Chang Tsuei and John Kirtley and, independently, Dale van Harlingen and coworkers—that led to such a conclusion. Yamada also claims that, “from the theoretical point of view the superconductivity in strongly correlated electron systems can be discussed on the basis of the Dyson—Gorkov equation which corresponds to the Eliashberg equation based on the vertex function arising from the electron—electron interactions in place of the usual electron—phonon interactions.” Does the Migdal theorem apply and thus justify such a conclusion? Yamada hardly discusses this point, which is the cornerstone for the applicability of the Eliashberg theory.

Admittedly, it is difficult, though not impossible, to write a book on topics such as heavy fermions and high-*T* _{c} superconductivity, topics that are still in a state of development. *Electron Correlation in Metals* contains detailed calculations but not enough discussion on the meaning of those calculations. In what regimes are the given approaches valid? What valuable explanation about the data can such methods give? What are the limitations? What are some of the other promising approaches? A thorough examination along those lines would have been desirable and valuable.

I was initially happy to see that some of the fascinating new developments of high-temperature superconductors, organic superconductors, heavy fermions, and unconventional superconductivity in the ruthenates were making their way into a textbook. The book does fill a gap by introducing readers to a variety of more recent topics in correlated-electron systems. Alexander Hewson’s *The Kondo Problem to Heavy Fermions * (Cambridge U. Press, 1993) essentially covers the same material found in Yamada’s first four chapters, but with greater emphasis on experiments and with in-depth discussion of scaling and renormalization-group techniques, exact solutions, and 1/*N* methods. Yamada sacrifices those topics to discuss some newer ones instead. Yet overall I was disappointed by the depth, or lack thereof, of the author’s treatment.

*Electron Correlation in Metals* could serve as a supplementary reference in a graduate-level, special-topics course in condensed matter theory. When confronting a new problem, it is often instructive to take a first stab at it using perturbative techniques—and for that, Yamada is indeed an expert. But thereafter, one must take a dispassionate look at the experimental data to ascertain whether the theory actually works. On that score, the reader would have to complement Yamada’s book with review articles and texts that examine more modern non-perturbative techniques, phenomenology, experiments—and, especially, open questions.