Boulevard of Broken Symmetries: Effective Field Theories of Condensed Matter , Adriaan M. J. Schakel , World Scientific, Hackensack, NJ, 2008. $79.00 (390 pp.). ISBN 978-981-281-390-9
These days interest in many-body condensed-matter physics is at an all-time high. New high-Tc superconductors continue to be discovered, the field has attracted the interest of the atomic-gas community, and possible links between quantum gravity and condensed-matter theory are generating growing fascination. Therefore, the demand for a concise, advanced-level introduction to the field theory of many-body physics has never been so great, especially given the scarcity of such textbooks.
I was thus delighted to learn of Adriaan Schakel’s text, Boulevard of Broken Symmetries: Effective Field Theories of Condensed Matter. As he states in the introduction, the book is a bold attempt at a self-contained presentation “free of historic constraints” (page viii). Schakel develops a pedagogical approach to many-body physics that is based on path integrals; he avoids the traditional use of operators and wave-functions until the final chapters. That expedient procedure enables the book to cover far more ground than typically possible given its size.
A few comparisons with existing texts may be useful to the potential reader. Alexander Altland and Ben Simon’s Condensed Matter Field Theory (Cambridge University Press, 2006) presents both operator and pathintegral approaches. John Negele and Henri Orland’s Quantum Many-Particle Systems (Westview Press, 1999) and Xiao Gan Wen’s Quantum Field Theory of Many-Body Systems (Oxford University Press, 2004) are examples in which the traditional operator approach is set aside. In its nearly exclusive focus on effective field theories from a path integral approach, Schakel’s book is closest to Wen’s.
Boulevard of Broken Symmetries is written in a style that will have particular appeal to students of statistical mechanics and field theory who wish to learn about interacting condensedmatter systems. The book’s breadth of coverage is also attractive, as it spans topics that include cold atomic gases, superconductivity, magnetism, and the quantum Hall effect. Schakel introduces new topics not found in most other texts—for example, the duality between vortices and particles in two-dimensional superfluids, the crossover from Bose–Einstein condensate to Bardeen-Cooper-Schrieffer state, homotopy theory, and composite fermions in the fractional quantum Hall effect.
Although my initial impressions of the book were extremely favorable, my enthusiasm was later tempered by reservations that arose after a more careful reading. At their heart is a concern that the pathintegral approach is not yet mature enough to replace the more traditional operator-wavefunction approach. Schakel seems to realize that limitation in the book’s later chapters, where his description of the quantum Hall effect is based almost entirely on the wavefunction approach. The book’s early chapters make no reference to operator-based quantum mechanics.
Forgoing that more standard approach might suit a reader wellversed in quantum field theory who is seeking an introduction to condensedmatter physics, but it leads to a rather difficult read for students with only a graduate-level knowledge of quantum mechanics. It’s a pity Schakel chose not to mention coherent states, which link quantum optics, operator- and wave-function-based quantum mechanics, and path integrals. That means the BCS wavefunction—the most famous coherent state—does not appear in the text.
Another serious shortcoming of the book lies in its treatment of linear response theory, a topic at the heart of condensed-matter physics. Schakel develops linear response theory in terms of real-time Feynman propagators, which are a mix of traditional retarded and advanced response functions. The student will find that combination very confusing.
Despite my reservations, I consider Boulevard of Broken Symmetries to be a courageous attempt to reformulate how to teach the field theory of condensedmatter physics. Even if it falls somewhat short of that lofty goal, its broad scope will certainly be attractive to those with a field-theory background who are seeking a rapid introduction to the bustling many-body physics sub-field of condensed matter.
