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In this lecture, we introduce the notion of adiabatic transport on the Brillouin zone as a tool to extract topological information about a Bloch Hamiltonian. We discuss adiabatic transport first in quantum mechanics and derive the Berry phase for an adiabatic variation of a parameter-dependent Hamiltonian. We then discuss the Berry phase in the context of cond-mat systems and relate it to the polarization of a one-dimensional systems.
The video lecture is divided into three parts, corresponding to the three sections in the lecture notes. You can also check out the video lectures at our Vimeo showcase.
This first section takes a detour away from cond-mat systems to derive the Berry phase for time-dependent quantum systems. We define Berry connection and curvature and discuss some of their important properties.
In this section, we look at the adibatic transport over the Brillouin zone. We also compute the Berry connection and curvature for some specific tight-binding models.
In this section, we discuss our modern understanding of polarization in one-dimensional systems, which is naturally expressed in terms of a Berry phase computed along loops in the Brillouin zone.
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