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This is the first of two lectures on the interplay between symmetry and topology in free fermionic systems. We start by a discussion of the implementation of symmetry transformation on single particle Hilbert space as unitary or antiunitary operators. We further discuss the former case, which includes all lattice symmetries, and show how the eigenvalues of these symmetry operators at high-symmetry points in the Brillouin zone can be used to diagnose topological phases.
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 discusses symmetry transformations as they apply to free fermionic systems, and their implementation as unitary or antiunitary operators.
We discuss the unitary symmetry transformation in more details, and derive the conditions on the Hamiltonian, represented in various forms, for such a transformation to be a symmetry.
We discuss the implementation of unitary symmetries for systems with multiple bands, where the transformation of internal degrees of freedom must be taken into account. We also discuss the labeling of bands using symmetry eigenvalues at high-symmetry points in the Brillouin zone and the use of these symmetry labels in diagnosing topological phases.
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