The central paradigm of modern condensed-matter physics is the notion of emergent behavior: Even though the underlying atomic-level details of condensed matter are extremely complicated, certain properties of the material—the emergent properties—can be captured by simple models with just a few parameters. If we can solve such a model exactly, or at least in a controlled way, then we can explore with confidence how its few parameters influence important characteristics of the system. By contrast, even if we could construct a fully realistic model that included all the microscopic complications, we would only be able to study the model by making uncontrolled approximations. Under those conditions we could hardly vouch for the model’s predictions.
The study of minimally complicated models is therefore central to the field of condensed-matter physics. Those models, and the tools needed to understand them, are the subject of Ramamurti Shankar’s new book, Quantum Field Theory and Condensed Matter: An Introduction. Shankar, a professor of physics at Yale University, is the author of the highly regarded textbook Principles of Quantum Mechanics (2nd edition, 1994), and his gift for clear exposition is once again on display in his latest work.
The first part of the text establishes the link between classical statistical mechanics, as represented by the Ising model, and quantum systems with many degrees of freedom. Shankar describes the link from both directions: The classical system’s transfer matrix maps to the quantum Hamiltonian, and Feynman’s quantum path-integral sum over trajectories maps to the classical sum over configurations. In each case one of the classical space directions corresponds to imaginary time evolution in the quantum realm.
With those ideas established through concrete and detailed examples, Shankar turns to the Wilsonian renormalization group. He explains how the long correlation lengths in classical second-order phase transitions both explain and are explained by the necessity of renormalization in a continuum field theory. The reader sees how some microscopic parameters become irrelevant at the macroscopic scale and why in the end only a few parameters control the emergent behavior.
In the second part of the text, Shankar applies the tools acquired earlier to such model systems as nonrelativistic electrons in one and two dimensions, where the classical ideas of Landau–Fermi liquid theory fail. He uses bosonization to solve the famous Luttinger model and to explain the Kosterlitz–Thouless transition in the XY model. The last chapter discusses the quantum Hall effect and finishes with a short introduction to Chern–Simons topological field theory. The text includes exercises that invite the reader to explore further and that fill in some of the omitted expository steps.
Several books on the market have titles similar to Shankar’s, including Alexander Altland and Ben Simons’s Condensed Matter Field Theory (2nd edition, 2010), Eduardo Fradkin’s Field Theories of Condensed Matter Physics (2nd edition, 2013), and the venerable Quantum Theory of Many-Particle Systems by Alexander Fetter and John Walecka, now available from Dover (2003).
What is different about Shankar’s text? For one thing, it is shorter. Quantum Field Theory and Condensed Matter is just 450 pages long, compared with 640 pages for Fetter and Walecka and around 800 pages for each of the other two texts. Accordingly, Shankar’s book is less ambitious in its aim and more selective in its content. That makes it both a more introductory text and a friendlier read. It will be a good textbook for a one-semester first-year graduate course.
Michael Stone obtained his PhD in particle physics in 1976. He is the editor of reprint editions of Quantum Hall Effect (1992) and Bosonization (1994), and the author of Physics of Quantum Fields (2000) and, with Paul Goldbart, Mathematics for Physics: A Guided Tour for Graduate Students (2009).