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This is the second of two lectures on the interplay between symmetry and topology in free fermionic systems. In this lecture, we discuss the antiunitary symmetries of the single particle Hilbert space, in particular the time-reversal and particle-hole symmetries. We discuss the time-reversal symmetry in detail and introduce the quantum spin Hall insulator as the first example of a symmetry-protected topological phase. We also show how particle-hole symmetry can be implemented antiunitarily on the Nambu space, a single particle Hilbert space formed by the direct sum of the single-particle and single-hole Hilbert spaces.
The video lecture is divided into four parts. You can also check out the video lectures at our Vimeo showcase.
This first section consists of a discussion on some general properties of antiunitary symmetries and their representation as a combination of a unitary symmetry and a "complex conjugation" acting on the Hilbert space.
In this section, we look at the time-reversal symmetry more closely, including its action on spin and on some of the previously studied tight-binding models.
In this section, we introduce the Bernevig-Hughes-Zhang model using two Chern insulators with opposite Chern number coupled together. This system exhibts a quantum spin Hall insulator — a topological insulator phase protected by the time-reversal symmetry.
In this section, we introduce particle-hole conjugation as it acts on the many-body Hilbert space, and show how it can be represented by an antiunitary operator on the Nambu space, consisting of single-particle and single-hole states.
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