Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Part One One-Electron Theory
- Part Two Topological Phases
- 8 Topological Aspects of Condensed Matter Physics: A Historical Perspective
- 9 Topological Preliminaries
- 10 Berry-ology
- 11 Topological Aspects of Insulator Band Structure and Early Discoveries
- 12 Dirac Materials and Dirac Fermions
- Part Three Many-Body Physics
- References
- Index
11 - Topological Aspects of Insulator Band Structure and Early Discoveries
from Part Two - Topological Phases
Published online by Cambridge University Press: 06 March 2020
- Frontmatter
- Dedication
- Contents
- Preface
- Part One One-Electron Theory
- Part Two Topological Phases
- 8 Topological Aspects of Condensed Matter Physics: A Historical Perspective
- 9 Topological Preliminaries
- 10 Berry-ology
- 11 Topological Aspects of Insulator Band Structure and Early Discoveries
- 12 Dirac Materials and Dirac Fermions
- Part Three Many-Body Physics
- References
- Index
Summary
Delineates how the ideas of topological equivalence and adiabatic continuity lead to the emergence of distinct classes of insulator Hamiltonians, and how this, in turn, leads to bulk-boundary correspondence – the connection between bulk topological invariants and edge or surface states. Classification of topologically nontrivial and trivial phases, based on fundamental discrete symmetries and dimensionality, the “tenfold way," is explained. Mapping of d-dimensional Brillouin zones onto a d-dimensional Brillouin torus and Bloch Hamiltonians are defined. and construction of Bloch bundles on the torus base manifold is outlined. Time-reversal symmetry, Kramers’ band degeneracy, “time-reversal invariant momenta,” and the implied vanishing of Berry’s curvature are delineated. The integer quantum Hall effect and the modern theory of polarization are discussed in detail. Z2 topological invariant is derived using the sewing matrix, time-reversal polarization and the non-Abelian Berry connection.
- Type
- Chapter
- Information
- Advanced Quantum Condensed Matter PhysicsOne-Body, Many-Body, and Topological Perspectives, pp. 286 - 330Publisher: Cambridge University PressPrint publication year: 2020