The energy bands in the momentum space of crystalline matter are characterized by topological invariants, which cannot be altered as long as the energy gap between the bands remains open. Such a description applies both to fully gapped insulators and superconductors, as well as to various types of band degeneracies (so-called “nodes”) in semimetals and metals. Topological invariants are usually captured by Abelian groups, such as the Kane-Mele Z2 invariant of time-reversal-symmetric topological insulators, or the integer Chern number that stabilizes Weyl nodes.

In this seminar, I will discuss our recent discovery of a non-Abelian topological invariant that arises in certain space-time-inversion symmetric crystals [1,2]. I will first explain the origin and the geometric meaning of its non-commutative behavior, which suggests the possibility of non-trivial braiding of band nodes inside the momentum space. Afterwards I will consider the interplay of the non-Abelian topology with crystalline symmetries, and show how it influences various species of band nodes, including: chain nodes in elemental scandium [1], Weyl nodes in ZrTe [3], and triple nodal points in Li2NaN [4].

References:

[1] Q.S. Wu, A. A. Soluyanov, and T. Bzdušek, Non-Abelian band topology of non-interacting metals, Science 365, 1273--1277 (2019)

[2] Apoorv Tiwari and Tomáš Bzdušek, Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101, 195130 (2020)

[3] A. Bouhon, Q.S. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nature Physics 16, 1137—1143 (2020)

[4] P. M. Lenggenhager, X. Liu, S. S. Tsirkin, T. Neupert, and T. Bzdušek, From triple-point materials to multiband nodal links, Phys. Rev. B 103, L121101 (2021)