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The bulk-boundary correspondence ― a relationship between the (bulk) topological invariant of a gapped system and the existence of topologically protected states localized on its boundaries ― is a remarkable feature of topological phases. In this lecture, we discuss the boundary modes, first in a continuum model described by a Dirac Hamiltonian, where one can show the existence of localized modes under rather general conditions, and then for various lattice models. We also discuss charge transport for a Hamiltonian undergoing an adiabatic periodic process and to relate the Chern number and the boundary modes to the amount of charge transferred in such a process, thereby motivating the bulk-boundary correspondence.
The video lecture is divided into three parts. You can also check out the video lectures at our Vimeo showcase.
In the section, we show that the Dirac equation with a mass domain wall ― a spatial region where the "mass" flips sign ― always contains a fermionic mode localized on this domain wall. We use this result to study the edge states for topological insulators in two dimensions.
In this section, we look at the boundary modes for 1d lattice models. In particular, we obtain the localized Majorana modes at the end of Kitaev chains.
In this section, we study the Thouless charge pump and show that for 1d topological insulators, under a periodic process, the net charge transferred is related to a Chern number. We also relate the Chern number to Hall conductivity for 2d systems.
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