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Topological insulators and topological superconductors. (English) Zbl 1269.82001
Princeton, NJ: Princeton University Press (ISBN 978-0-691-15175-5/hbk; 978-1-400-84673-3/ebook). ix, 247 p. (2013).
Topological insulators are new states of quantum matter which can not be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. This volume provides a comprehensive introduction to the theory of topological insulators and superconductors, which is an important field in condensed matter physics. The book presents an array of increasingly complicated problems centered around the idea of the topology of a band in \(k\)-space. A central role is played by the basic theorems asserting that the Chern number determines the Hall effect.
The book begins with simple concepts such as Berry phases, Dirac fermions, Hall conductance and its link to topology, and the Hofstadter problem of lattice electrons in a magnetic field. It moves on to explain topological phases of matter such as Chern insulators, two- and three-dimensional topological insulators, and Majorana \(p\)-wave wires. The volume under review also covers zero-modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field theory and topological indices. The book analyzes recent topics in condensed matter theory and concludes by surveying active subfields of research such as insulators with point-group symmetries and the stability of topological semimetals. Problems at the end of each chapter offer opportunities to test knowledge and engage with frontier research issues. The content of the book is divided into 18 chapters. An appendix devoted to 3D topological insulators in a magnetic field may be also found in this volume.
The book under review may be extremely useful to both graduate students and more senior researchers.

82-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics
00A79 Physics
82D37 Statistical mechanics of semiconductors
82D55 Statistical mechanics of superconductors
81T45 Topological field theories in quantum mechanics