Topological Quantum Chemistry Becomes Predictive

by Tommy on 5/05/2017

Wallpaper Fermions and the Topological Dirac Insulator, Benjamin J. Wieder, Barry Bradlyn, Zhijun Wang, Jennifer Cano, Youngkuk Kim, Hyeong-Seok D. Kim, A. M. Rappe, C. L. Kane and B. Andrei Bernevig, Submitted on March 21, 2017 (3 May 2017)

Recent developments in the relationship between bulk topology and surface crystalline symmetries have led to the discovery of materials whose gapless surface states are protected by crystal symmetries. In fact, there exists only a very limited set of possible surface crystal symmetries, captured by the 17 “wallpaper groups.” We show that all possible crystalline insulators, symmorphic and nonsymmorphic, can be exhaustively characterized by considering these groups. In particular, the two wallpaper groups with multiple glide lines, pgg and p4g, allow for a new topological insulating phase, whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion. Like the surface state of a conventional topological insulator, the surface Dirac fermion in this “topological Dirac insulator” provides a theoretical exception to a fermion doubling theorem. Unlike the surface state of a conventional topological insulator, it can be gapped into topologically distinct surface regions while keeping time-reversal symmetry, allowing for networks of topological surface quantum spin Hall domain walls. We report the theoretical discovery of new topological crystalline phases in the A2B3 family of materials in SG 127, finding that Sr2Pb3 hosts this new topological surface Dirac fermion. Furthermore, (100)-strained Au2Y3 and Hg2Sr3 host related topological surface hourglass fermions. We also report the presence of this new topological hourglass phase in Ba5In2Sb6 in SG 55. For orthorhombic space groups with two glides, we catalog all possible bulk topological phases by a consideration of the allowed non-abelian Wilson loop connectivities, and we develop topological invariants for these systems. Finally, we show how in a particular limit, these crystalline phases reduce to copies of the Su-Schrieffer-Heeger model.

See also:

Topological quantum chemistry, Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo and B. Andrei Bernevig (6 March 2017)

The past decade’s apparent success in predicting and experimentally discovering distinct classes of topological insulators (TIs) and semimetals masks a fundamental shortcoming: out of 200,000 stoichiometric compounds extant in material databases, only several hundred of them – a set essentially of measure zero – are topologically nontrivial. Are TIs that esoteric, or does this reflect a fundamental problem with the current piecemeal approach to finding them? Two fundamental shortcomings of the current approach are: the focus on delocalized Bloch wavefunctions – rather than the local, chemical bonding in materials – and a classification scheme based on a collection of seemingly unrelated topological indices. To remedy these issues, we propose a new and complete electronic band theory that assembles the last missing piece – the link between topology and local chemical bonding – with the conventional band theory of electrons. Topological Quantum Chemistry is a description of the universal global properties of all possible band structures and materials comprised of a graph theoretical description of momentum space and a dual group theoretical description in real space. This patches together local k⋅p dispersions into distinct global groups of energy bands: we classify all the possible bands for all 230 crystal symmetry groups involving s,p or d orbitals on any of the Wyckoff positions of every space group. We show how our topological band theory sheds new light on known TIs, and demonstrate the power of our method to predict a plethora of new TIs.

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