A synthesis of physical connectedness

Topology is the study of connectedness. It addresses the question: Given two objects, can one be smoothly deformed into the other? Two objects that do not fit that criterion are said to be topologically distinct. Think of an ordinary rubber band. It cannot be stretched into a Möbius strip unless it is cut, twisted, and glued back together. Topology has broad applications in the physical sciences, and perhaps the most well-known is the theory of topological defects: large-scale structures such as vortices and domain walls that describe stable, twisted configurations of classical fields that cannot be deformed away.

A more subtle application of topology began to emerge in the 1980s with the discovery of the quantum Hall effect. Topologically speaking, it arises from the twisting not of classical fields but of electronic wavefunctions, or of collections of them that are called bands. Remarkably, the degree of their twisting—a topological invariant called the Hall conductivity—is characterized by an integer that can be directly measured in units of e²/h, where e is the electron charge and h is Planck’s constant. In other words, a combination of fundamental constants defines a quantum of conductance.

The subsequent discovery of the fractional quantum Hall effect, in which the Hall conductivity was found to be a rational fraction of e²/h, led to the notions that many-body wavefunctions could be twisted and that quasiparticles may exist and behave like fractions of electrons. Those eventually led to the modern concept of topological order, which provides a way to organize phases of matter that is complementary to the conventional one based on symmetry and order parameters.

Although they were appreciated for their beauty, topological applications in physics remained largely limited to the esoteric realm of ultrahigh magnetic fields in artificial structures until the 2000s, when it was discovered that the concepts applied much more generally. Today they are the bread and butter of theoretical condensed-matter physics. There are now databases of topological characteristics of materials, corporate efforts to build topological quantum computers, and much more. The theory is much more evolved, and its mathematical treatment is more sophisticated and has been applied to such topics as band structure, exotic quantum magnets, and unconventional superconductors.

Roderich Moessner and Joel Moore’s recent book, Topological Phases of Matter, provides a synthesis of the vast subject. The authors begin with an introduction before delving into a compact summary of background material including the Berry phase, Landau levels, tight-binding models, and Landau theory. They also include a succinct section on the mathematics of topology, which readers will benefit from because it summarizes material often found in disparate sources in one place. Subsequent chapters discuss the integer quantum Hall effect, quantum pumps, and the Chern number; examples of fractionalization including the fractional quantum Hall effect, spin liquids, gauge theories, and topology in conductors and superconductors; and even topological ideas in quantum computing and nonequilibrium-driven systems.

It became clear to me only after reading the book just how ambitious and challenging it is to cover all that material. Moessner and Moore generally chose to illustrate through examples instead of presenting a systematic overview of the topic. That makes the book more accessible, but at the same time it can be limiting. Moreover, the choice of examples may sometimes reflect the authors’ predispositions rather than being entirely representative. I found it strange, for example, that although disorder and localization are discussed in chapter 8, there’s no discussion of them when the integer quantum Hall effect is introduced in chapter 3, nor any mention of the Kubo formula. Both are essential to understanding the exactness of quantization and the existence of plateaus in the Hall effect.

Analogously, the chapter on gauge theory contains a thorough yet concise discussion of Ising gauge theory, but it presents without explanation Alexander Polyakov’s famous result that proved the instability of U(1) gauge theory in two spatial dimensions. I also was frustrated that the presentation of spin liquids in several chapters completely avoids the concepts of Gutzwiller projection and general parton constructions, both of which are central to the subject. Finally, tensor networks and matrix product states, which provide a powerful tool to understand many topological phases, are essentially absent.

But Moessner and Moore are clear about the choices they make, and in the preface, they explain their reasons for being less than encyclopedic. On balance, Topological Phases of Matter is an enjoyable read and a particularly broad introduction to diverse aspects of topological ideas in quantum theory. Its main weaknesses are ones of omission: Many results aren’t derived in the text, and some topics—such as those mentioned above—are left out entirely. That means that the level of detail is probably less than one would want to present in a graduate physics course, but a knowledgeable instructor could fill in the gaps and rely on the book as a resource to guide the order of topics presented in the class. The volume should be a helpful resource for anyone wanting to learn more about a topic that has become increasingly central to modern physics.

Leon Balents is a professor of physics and a permanent member of the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. His research focuses on the interaction of quantum particles.