The goal of quantum many-body physics is to understand the emergent properties—probed by thermodynamic, spectroscopic, and linear response functions—of a system of many interacting particles. As early as the 1950s, the methods of quantum field theory were applied to quantum fluids of fermions and bosons. Those efforts culminated in 1957 in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. In 1963 Alexei Abrikosov, Lev Gor’kov, and Igor Dzyaloshinskii wrote their classic book *Methods of Quantum Field Theory in Statistical Physics* (Prentice-Hall) on the use of Feynman diagrams to attack many-body problems at finite temperature. Terse but full of insights, the book had an enormous impact and is still used by practitioners.

However, the field has advanced tremendously since the 1960s. Subsequent decades saw great progress in addressing new many-body systems—such as those exhibiting the Kondo effect, disordered systems, superfluid helium-3, and unconventional superconductors—and the development of new tools, such as functional integrals and the renormalization group. Students and instructors of quantum many-body physics need an updated, modern textbook that covers those developments. *Introduction to Many-Body Physics*, a new book by Piers Coleman, successfully fills the need. Coleman, an eminent condensed-matter theorist at Rutgers University, covers his subject with pedagogical flair and attention to detail.

The book, which assumes the reader has no more than a knowledge of first-year graduate quantum mechanics, has two central aims. The first is to introduce a variety of techniques in many-body physics. The second is to illustrate the power of these techniques using detailed examples from condensed-matter physics.

Coleman seems keenly aware of a common problem with many-body textbooks: Readers, especially newcomers to the field, tend to get overwhelmed by the mathematics and lose sight of important physical ideas. To ensure that does not happen, he discusses the phenomenology and experimental background for each topic and even includes a bit of history. He also includes plots of experimental data that serve to motivate the theory or illustrate tests of theoretical predictions. Coleman makes excellent use of color illustrations to illuminate qualitative ideas and, at regular intervals, emphasizes important results by highlighting them in boxes or tables. Experienced researchers might find some of the derivations overly lengthy, but beginning graduate students will benefit greatly from the level of detail. Students will also find the many solved examples and exercise problems useful.

*Introduction to Many-Body Physics* is more than 800 pages long and thus is able to provide wide-ranging coverage of topics central to the field. The book skillfully matches theoretical concepts with experimental probes. For example, the chapter on linear response successfully emphasizes the connection between theoretical correlation functions and observable consequences like photoemission, neutron scattering, and optical conductivity.

Two chapters in the middle of the book develop key phenomenological ideas. One of them focuses on Ginzburg–Landau theory and does a marvelous job of introducing the ideas of broken symmetry, phase rigidity, topological defects, and the Anderson–Higgs mechanism. The other carefully explains phenomenological aspects of Landau’s Fermi liquid theory, but has little discussion of its microscopic formulation, which would have been of great value for students. Furthermore, the discussion of the *T*^{2} resistivity arising from electron–electron interactions is somewhat misleading. The reader should have been warned that this mechanism by itself does not give rise to resistance, unless *umklapp* scattering or disorder degrades the momentum of electrons.

Later chapters of the book describe several topics in condensed-matter physics. The first is itinerant magnetism, which Coleman uses to illustrate an application of functional integrals. Next come two chapters on superconductivity that cover weak coupling BCS theory, retardation effects, *p*-wave pairing in ^{3}He, and *d*-wave pairing in lattice models.

The final three chapters focus on the problems of local moments in metals and of heavy-fermion physics. Coleman has made pioneering contributions to those topics, and readers can gain much insight here. He even discusses the hotly debated possibility of topological Kondo insulator behavior in samarium hexaboride. However, the discussion of topological insulators is much too brief and readers will have to turn elsewhere for a deeper understanding of topological states of quantum matter.

Despite its length, *Introduction to Many-Body Physics* does not cover several topics that could be included in a modern many-body physics course. Among them are the Bogolyubov Bose gas, the superfluid-to-Mott insulator transition in the Bose–Hubbard model, and the BCS-to-BEC (Bose–Einstein condensation) crossover—all problems of great relevance to ultracold-atom experiments. Nor does Coleman discuss one-dimensional systems, the fractional quantum Hall effect, or numerical methods for many-body problems, all of which rightly deserve books of their own.

Coleman concludes *Introduction to Many-Body Physics* with a short but insightful epilogue in which he summarizes challenges for the future, including many-body problems that cannot be understood within the Landau paradigms of broken symmetry and well-defined quasiparticles with conventional quantum numbers. A reader who has mastered the material in this excellent book should be in a strong position to take on problems that have resisted conventional solutions.

**Mohit Randeria** is a professor of physics at the Ohio State University. He is a condensed matter theorist who works on quantum materials and ultracold atoms.