The fields of Physics and Chemistry approach electronic band structure differently. Physicists develop an understanding of bands in momentum space, while chemists look at the bonding of orbitals in real space. For years, electronic band theory has been dominated by important quantitative aspects such as the relative energy of bands in materials composed of chemical elements sitting on lattice sites. In the 1980s and in the early 2000s Joshua Zak, and then Louis Michel, Henri Bacry and Zak made the huge, but not well-known, leap of understanding the qualitative connection between real space and momentum space. For the classical (spinless) groups, they defined the connection between orbitals of elements sitting in lattice positions and bands in momentum space. They taught us to think of a band as a representation and showed that all bands fall in blocks akin to “irreducible representations” in group theory, called Elementary Band Representations (EBRs); they also presented proofs that EBRs are connected in the Brillouin zone — they cannot be decomposed in smaller sub-bands. In this colloquium, we present work that first completes Zak’s idea by extending it to the double (spinful) groups in the presence of time-reversal. We identify what lattice positions and what orbital representations are “special” and classify all the (~10,000) elementary band representations possible in the 230 space single and double groups, with and without Time-Reversal symmetry. These are now tabulated on the Bilbao Crystallographic Server, the premier crystallography website. We then move beyond the work of Zak and collaborators by bringing topological insulators and topological semimetals into the theory of band representations. Crucially, we realize that the Elementary Band Representations, the irreducible representations of band theory, can actually be “reduced” or decomposed — different from the initial prediction; however, when they decompose, they necessarily become topological. We present a mapping of bands in the Brillouin zone to graph theory that allows us to compute whether an EBR is connected or disconnected, and classify all the EBRs existent in nature — also now tabulated on the Bilbao Crystallographic Server. Many new classes of topological insulators and semimetals arise in this way. Fundamentally, this theory offers a link from real space orbitals to topological physics, allowing for the real-space design of several new types of topological materials, which we also present.
Hamilton Colloquium Series: B. Andrei Bernevig, Princeton University, "Topological Quantum Chemistry"
Thu, Nov 9, 2017, 4:00 pm
A free lecture open to the public.