2022-04-28 Seminar Report

In this lecture series, Professor Ozawa gave an introduction to topological insulators which are materials whose wavefunctions show nontrivial topological structures in momentum space. In the first two lectures, he introduced the so-called Su-Schrieffer-Heeger model and the bulk-edge correspondence which links edge states with winding number of certain operator in the Hamitonian. Such correspondence has its origin in mathematics called Toeplitz Index theorem which is a special case of Atiyah-Singer index theorem.

In the last two lectures, Chern insulators and quantum metrics are introduced. Eigenvectors of the Hamiltonian define a map from the momentum space (typically a torus) to a complex projective space. The pullback Fubini-Study metric (and standard Kahler form) defines the so-called quantum metric (and Berry curvature) on the momentum space. Using Chern-Weil theory, Chern classes/characters are then defined. Chern classes which are originally notions from differential geometry/topology play an important role in topological insulators. A necessary and sufficient condition is also given when the above mentioned map is an immersion and realises momentum space as a Kahler submanifold of the projective space.

Reported by Yalong Cao

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