This textbook introduces topological defects and solitons at a level suitable for advanced undergraduates and beginning graduate students in physics and materials science. It avoids the formal mathematics of topology, and instead concentrates on the physical properties of these topological structures.

The first half of the book concentrates on fundamental principles of defects and solitons, and illustrates these principles with a single example—the xy model for 2D magnetic order. It begins by defining the concept of a winding number, and uses this concept to describe the topology of defects (vortices or disclinations) and solitons (domain walls), carefully identifying the similarities and differences between these two types of topological structures. It then goes on to discuss physical properties of defects and solitons, including free energy, dynamics, statistical mechanics, and coupling with curvature. It shows how these concepts emerge from a theory with variable magnitude of order, and hence how topology can be viewed as an approximation to physics.

The second half goes on to explore a wider range of topological defects and solitons. First, it considers more complex types of order—2D nematic liquid crystals, 3D magnetic or liquid-crystal order, 2D or 3D crystalline solids—and shows how each type of order leads to specific topological structures. Next, it discusses defects and solitons that are characterized by 2D or 3D measuring surfaces, not just 1D loops, including hedgehogs, skyrmions, and hopfions. These structures are more complex, but they can still be understood using the same fundamental principles. A final chapter describes the formation of phases with regular arrays of defects or solitons.

“Notwithstanding their name, defects turn out to be, as Selinger’s book shows, very useful in a variety of fields, particularly in the study of liquid crystals, magnets, and other ordered matter systems. The topic has become increasingly important, as shown by the more than twofold increase in the number of papers dealing with topological defects listed by Web of Science in the last ten years, making the book certainly timely. Selinger has managed to introduce the subject in a way that is accessible to a wide audience of students and researchers, starting with simple physical examples, relatively straightforward maths, and helpful illustrations to arrive at complex and often unfamiliar concepts, such as Skyrmions and Hopfions. I appreciate the book's pragmatic yet elegant approach and have no hesitation in recommending it.” (Claudio Zannoni, Emeritus Professor of Physical Chemistry, University of Bologna, Italy)

“In this elegant, concise book, Jonathan Selinger allows readers to reap insights gained during a distinguished career focused on understanding liquid crystals, magnets and other complex materials.  With a minimum of fancy mathematics and a maximum of physical intuition (rendered in part through beautiful illustrations), he leads the reader with lucid prose to a basic understanding of the essential features of topological defects and solitons in condensed matter physics.” (David R. Nelson, Arthur K. Solomon Professor, Harvard University)

“In his book Introduction to Topological Defects and Solitons Jonathan Selinger brings the reader from the basic notions concerning topological defects and solitons all the way to the more sophisticated ones of “hedgehogs,” “skirmions,” “hopfions,” etc. They are exposed with clarity, avoiding unnecessary difficulties, but pointing at all aspects of the fascinating domain of topological singularities and solitons in magnetic systems, liquid crystals and also superconductors on occasion. Upon reading the book, one has the impression of listening to the master explaining at the blackboard the concepts with the use of well-designed drawings and well-chosen words. Without any doubts, this book will be useful to our community and beyond.” (Jacques Prost, Institut Curie, France)