Quantum Theory of the Electron Liquid Gabriele F. Giuliani and Giovanni Vignale , Cambridge U. Press, New York, 2005. $95.00 (777 pp.). ISBN 0-521-82112-6

* Quantum Theory of the Electron Liquid * is a veritable encyclopedia of a field that continuously rejuvenates itself with fresh physics discoveries and novel materials. Nothing escapes the attention of authors Gabriele F. Giuliani and Giovanni Vignale, who seem to cover the gamut. Topics range from Eugene Wigner’s 1934 theory of the three-dimensional electron crystal to the most recent developments of the time-dependent density functional theory, from weakly interacting electrons in two and three dimensions to the novel correlation phenomena occurring in a 1D conductor or in ultra-high magnetic fields.

The book, unquestionably attractive to the more experienced reader, is accessible and designed to be comprehensible to graduate students in condensed matter theory. It is an invaluable source of material for many-body theory as a part of condensed matter physics graduate courses. The reader is guided effectively through a large body of knowledge that the condensed matter community has produced over the course of 70 years.

Normally a book in this field would focus on a subclass of problems—strong correlations, density functional methods, lattices, 1D systems, and so forth. Instead, in 10 chapters, the authors cover with comparable care, clarity, and depth a broader area of the quantum theory of electronic systems. Topics are presented in a highly original manner, with the authors often deriving the same results through several different approaches.

Readers will likely find something remarkable just by opening the book at random. For instance, the authors provide the proof of Albert Overhauser’s famous Hartree–Fock instability theorem. Except for the original articles, I know of no other literature that gives the proof. Giuliani and Vignale uniquely spell out the validity limits of the celebrated random-phase approximation of David Bohm and David Pines, and they clearly show how that random-phase approximation can be improved. The book contains sophisticated and important concepts, such as spin-dependent effective electronic interactions, a subject virtually impossible to find in other textbooks.

Giuliani and Vignale cover density functional theory and allow ample space for the most recent developments in the time-dependent theory. In particular, they discuss in great detail the formalism of current-density functional theory in magnetic fields and the problem of the ultra-nonlocality of the exchange-correlation potential. Their formulation allows for the treatment of important effects—such as the intrinsic width of the collective excitations due to electron–electron interactions—that are inaccessible in a simple adiabatic approximation. The authors also discuss with authority the state-of-the-art microscopic theory for Landau parameters in a normal Fermi liquid. Recently, Fermi liquid and non-Fermi liquid theories have attracted considerable attention because of phenomena in the 2D, low-density electron liquid in semiconductor heterojunctions, where the role of interactions is particularly important. Readers can also find, for the first time in a textbook that I have seen, the renormalized Hamiltonian approach, a transparent method to obtain an approximate quasiparticle effective Hamiltonian.

The book’s overall treatment of the above topics is in some sense to be expected; the authors are leaders in the field and have contributed substantially to it. However, the book contains much more. It also offers a full chapter on the Luttinger-liquid theory, with diligent coverage of Duncan Haldane’s bosonization approach. All the main pillars of that approach and its physical consequences are explained in fewer than 50 pages. Similarly, for readers who want to learn or refresh their knowledge of the most exotic behaviors encountered in electron fluids, such as the quantum Hall effect, fractionally charged quasiparticles, or composite fermions, Giuliani and Vignale beautifully spell out those concepts in a self-contained form.

* Quantum Theory of the Electron Liquid * is enriched with 25 mathematical appendices elaborating on technical details. The appendices, coupled with the many exercises at the end of each chapter, make the book an unparalleled reference. Every graduate student and researcher studying condensed matter theory should obtain a copy of this exceptional text.