Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems , A. J. Leggett , Oxford U. Press, New York, 2006. $70.00 (388 pp.). ISBN 978-0-19-8526438
In Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems, Anthony Leggett has attempted to produce a graduate textbook for the Oxford University Press series on the various examples of macroscopic quantum-coherent states in condensed-matter physics. Chapters are devoted to liquid helium-4, Bose–Einstein condensation (BEC) in cold atoms, classical superconductivity, liquid helium-3, and cuprate superconductivity; a final, eighth chapter covers assorted exotic systems.
Not that Leggett, a 2003 Nobel laureate in physics and MacArthur Professor at the University of Illinois at Urbana-Champaign, would ever use the words “macroscopic quantum coherence”: He devotes much of the first two introductory chapters to explaining why he rejects the now-classic description in terms of coherent states and favors instead the narrower idea of off-diagonal long-range order. In so doing, he explicitly rejects the connection of coherence with broken gauge symmetry that eventually led to the electroweak theory; but more seriously, the words “Goldstone boson” do not appear in reference to the phonons and collective modes in helium, nor does the question of why they do not appear in superconductors. Because the Bardeen-Cooper-Schrieffer (BCS) theory has always been viewed as the poster child for the concept of broken symmetry, Leggett's refusal to look in those directions warrants questioning—and all the more in a work of reference.
The early chapters of the book also divagate on a number of questions that seem out of place in a text at this level. For instance, is it at all relevant that “there is no proof of the existence of Bose–Einstein condensation in any physical system” (page 46). If there were “proof” of nonexistence, it would be the premises of the proof that would be questioned, not the physics of liquid helium.
For graduate students who want a thorough grounding in some of the most fundamental aspects of quantum fluids—such as statistical mechanics in a rotating container, the Landau–Silin approach to metals, the dynamical theory of the dielectric constant of metals, and the theory of Feshbach resonances in dilute gases—Quantum Liquids would be very useful. And for those of us who have specialized in a particular branch of the field and need updating on the marvelous things that have been done with cold atoms or on the beautiful details of the liquid 3He story, the book is a wonderfully informative source. Each of the four chapters on the classic, well-understood cases of liquid 4He, Bose alkali gases, superconductivity, and liquid 3He is full of small gems of insight, typical of Leggett's finical style.
But from time to time it seems as if the author has distorted or ignored history. One wonders if Henry Hall and Joe Vinen would have been happy being dropped from the history of quantized vorticity in superfluid 4He. In the chapter on classical superconductivity, John Rowell is not mentioned in connection with the Josephson effect. And perhaps I might have earned some credit for the formalism in section 5.8, the one using time-reverse pairing. In addition, the entire and crucially important subject of flux lattices, flux pinning, and creep and flow is postponed to a cursory inclusion in the chapter on cuprates. The omission of the flux lattice and flow properties constitutes a serious incompleteness in a learning tool.
For my specialty, the cuprates, I had hoped to see a thoughtful, if idiosyncratic, treatment like those in the previous chapters; instead, the coverage in chapter 7 does not actually reflect the modern state of the subject. Again, Leggett's characteristic viewpoint comes into play: He says, in effect, that some may believe that the Hubbard model is useful—but he doesn't, offering no explanation (page 332). Yet Leggett is not finicky about the mathematically questionable basis of the antiferromagnetic spin-fluctuation theory. Another example: A long-ago paper by one of his close colleagues, Myron Salomon, shows that the transition is always of x–y character—that is, into a fluctuating paired liquid—which invalidates the naive Jeff Tallon phase diagrams that Leggett uses.
In a few paragraphs in chapter 8, Leggett dismisses the large field of organic superconductors. He does not mention established data that demonstrate the coincidence of a Mott transition in an organic material with that in an antiferromagnetic superconductor, nor does he mention the demonstrations of triplet superconductivity in organics. Instead, the subject is dismissed as “probably phonon-motivated” (page 352) despite the myriad evidences for Mott–Hubbard physics. Leggett at least admits that the heavy-fermion superconductors are likely to have exotic order parameters and an electronic mechanism, but his long-term love affair with antiferromagnetic spin fluctuations continues. Superconductivity in Sr2RuO4, whose order parameter so closely resembles that of his favorite 3He, is given a brief mention. Except for a final note on the BCS–BEC crossover, all of chapter 8 is cursory and out of date.
In summary, Quantum Liquids is a book that many condensed-matter theorists can read with profit. But because the text is so selective and occasionally misleading in parts, I would question its use as a comprehensive textbook for graduate students.
