Berry Phases in Electronic Structure Theory. Electric Polarization, Orbital Magnetization and Topological Insulators . By David Vanderbilt. Cambridge University Press, 2018. Hardback, pp. x+384. Price GBP 59.99. ISBN 9781107157651.

The book by David Vanderbilt, Berry Phases in Electronic Structure Theory, is a very pedagogical introduction to the role played by Berry phases in our understanding of the electronic properties of matter. It is indeed written by one of the prominent contributors to the field. Since their discovery in 1984, Berry phases have been used to understand or reinterpret a variety of phenomena such as charge pumping, polarization, orbital magnetization and Hall effects, as well as topological insulators. They are all discussed in the book, step by step, from an elementary level to a more detailed understanding. Throughout the book a Python package, PythTB, is used to provide numerical examples that support the theory given in the book. The code of the examples is given in the appendices so that it can be investigated by the reader. In the first chapter D. Vanderbilt gives an overview of the physics covered in the book, and explains on physical grounds several of the results obtained in later chapters. The second chapter reviews elementary concepts in solid-state physics. A rather lucid and self-contained discussion of density functional theory, Bloch functions and tightbinding Hamiltonians is given. The PythTB package is also introduced. This package, developed at Rutgers University, allows one to build and solve tight-binding models and compute Berry phases related properties. It is most useful to understand the concepts newly introduced in later chapters. From the mathematical point of view, the third chapter is the most important in the book. The Berry phase is first introduced as a Pancharatnam phase. A parameter space is considered, and the phase of a state vector is accumulated along a closed path which is covered in a finite number of steps. Later, the continuous limit is taken, and important concepts such as Berry connection and curvature are defined. The Chern theorem, which is that the integral of the Berry curvature on a 2-surface is quantized, is then stated. With regard to its importance for later applications, one may regret that a more mathematical derivation is not given, although convincing arguments are presented. After the discussion of Berry phases in general terms, several applications are considered. Adiabatic dynamics is reviewed, making apparent the role played by the Berry phase. Then, instead of considering a generic parameter space for a defined Berry phase, the Brillouin zone is chosen, and the parameters are the components of the electron wavevectors. This is used in most of the following chapters. Finally, the chapter is concluded with a discussion of Wannier functions. In particular, it is shown how the gauge freedom on the Bloch functions affects the average value of the position operator, the Wannier centres, and how they are related to Berry phases. This is very useful in later chapters to obtain a physical picture of the phenomenon under consideration. Chapters 4, 5 and 6 are devoted to applications of the general concepts developed so far. The modern theory of polarization is presented in Chapter 4, a theory developed by David Vanderbilt himself in the 1990s. In insulators, the textbook approach to polarization is based on a separation of the charge into free and bound charges, which then define polarizable entities. In solids, the separation is far from obvious, and this approach prevented practical computations for many years. The arguments given in the first chapter of the book, advocating for a multivalued polarization, are derived here. To ISSN 2053-2733