Notes on topological insulators

Abstract

This paper is a survey of the ${\mathbb{Z}}_2$-valued invariant of topological insulators used in condensed matter physics. The $\mathbb{Z}$-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The ${\mathbb{Z}}_2$ invariant is more mysterious, we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.

62 pages, 3 figures

References in corpus (10)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Field Theory of Time-Reversal Invariant Insulators
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• The bulk-edge correspondence for the quantum Hall effect in Kasparov theory
• Topological meaning of Z${}_2$ numbers in time reversal invariant systems
• A Non-Commutative Formula for the Isotropic Magneto-Electric Response
• Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid
• A Z${}_2$ index of Dirac operator with time reversal symmetry

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• Discrete Wigner-Weyl calculus for the finite lattice
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• A Two-Gauge Field Model for Magnetoelectric Boundaries
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• Kinematics and Dynamics of Quantum Walks in terms of Systems of Imprimitivity