Defects and Geometry in Condensed Matter Physics , David R. Nelson Cambridge U. Press, New York, 2002. $110.00, $40 paper (377 pp.). ISBN 0-521-80159-1 ISBN 0-521-00400-4 (paper)
For 25 years, David Nelson has made major contributions to the study of how defects influence the properties of such condensed matter systems as solids, liquid crystals, superfluids, and polymer solutions and melts. His dominance in the field has been reinforced by his excellent surveys in the proceedings of summer schools, workshops, and conferences from 1983 to 1996. Defects and Geometry in Condensed Matter Physics is primarily a compilation of eight of those reviews. Each article has a new short preface, which usually refers the reader to more recent reviews and provides updates of the bibliography. An opening chapter, which ties the collection together, makes clear that Nelson selected the reviews not to be comprehensive but to emphasize work in which he was directly involved. The book is aimed at graduate students in physics, physical chemistry, and chemical engineering, and to more advanced researchers in those fields.
Chapter 2, a reprint of Nelson’s review of defect-mediated phase transitions, was originally published in volume 7 (Academic Press, 1983) of the series, Phase Transitions and Critical Phenomena , edited by Cyril Domb and Joel Lebowitz. It is by far the most commonly cited of the eight reviews, and I find it the most accessible treatment of its subject. It is a masterly survey of the theory of vortex-driven transitions from superfluid to normal fluid in helium films and of melting two-dimensional solids. Early in his career, Nelson showed that if the melting of 2D solids was driven by the dissociation of dislocation pairs, it would lead not to a true liquid but to a “hexatic” phase, which has the flow properties of a fluid but the orientational order of a solid. This hexatic phase, a new sort of liquid-crystal phase, has a second continuous phase transition to an isotropic liquid. The transition is driven by translational line defects (dislocations) splitting into pairs of orientational line defects (disclinations). The only references in chapter 2 to work done since 1983 relate to experimental and computational verification of the hexatic phase. Chapter 6, based on the 1996 Les Houches, France, summer school proceedings, updates chapter 2 by connecting the original ideas with more recent applications to membranes and to vortices in superconductors.
The other chapters discuss various applications in detail. Two chapters on glasses and related systems deal with such problems as the competition between long-range and short-range order. For example, icosahedral order at small distances may lead to glassy disorder or it may lead to crystalline order as is seen in crystals of icosahedral viruses. A chapter on crumpled membranes, first published in 1988, contains the greatest number of recent references.
The discovery of high-temperature cuprate superconductors in 1986 raised many new questions about the behavior of topological defects in superconductors. Because of the weak coupling of the superconducting order between the cuprate planes, when magnetic flux penetrates the superconductor it creates vortices in the planes, but the paths connecting vortices in different planes have low energy per unit length. Two chapters discuss these questions and describe the different arrangements of the vortices—such as flux lattice, vortex glass, and vortex liquid—in cuprate superconductors. Those phases have all been identified in experimental work. A powerful tool here is the analogy between the possible paths of vortices from one cuprate plane to the next and the space-time trajectories of a set of bosons in two dimensions. The prefaces to these chapters contain references to more recent works, many with Nelson as an author.
The final chapter, on the statistical mechanics of directed polymers, relates particularly to polymer molecules that are constrained to be oriented in a particular direction, as in a nematic liquid crystal. The theory of such a polymer is analogous to the theory of the magnetic vortices in a superconductor. Both can be related to the quantum theory of bosons in two dimensions. A recent paper quoted in the preface showed hexatic order in condensed DNA molecules.
The book provides an admirable overview of Nelson’s achievements and of their relation to other work. All but one of the reviews are available in our library, but may not be available in smaller institutions. It is valuable to have all these contributions collected in one volume. I did not find the book easy reading, because often the articles concentrated on the details of what was then recent work, rather than on a broader perspective. There is also some repetition, and not quite enough linking material. I would have preferred a full-blown textbook giving the author’s mature perspective, but that is too much to expect from the chair of a major physics department who has a vigorous research program.
