(*last updated in 2020)

Band structures of non-interacting particles can be characterized by topological numbers; famous examples being the $\mathbb{Z}$-valued Chern number of Weyl points, and the $\mathbb{Z}_2$-valued Kane-Mele invariant of topological insulators. The groups capturing the topological invariant of single-particle energy bands were essentially always assumed to be Abelian – until we discovered a counterexample [1]. Led by intuition from the physics of biaxial nematic and cholesteric liquids, we showed that band nodes of certain $\mathcal{PT}$-symmetric and $C_2\mathcal{T}$-symmetric systems are characterized by a previously overlooked non-commutative quaternion charges.

The non-Abelian topological invariant implies, among other things, that certain species of nodal lines cannot be moved across one another by tuning the Hamiltonian parameters [1]. Remarkably, it also enables a “reciprocal braiding” effect, meaning that the capability of band nodes to pairwise annihilate depends on the trajectory used to bring them together in $\boldsymbol{k}$-space [2]. The non-Abelian band topology brings under one umbrella several notions of topological band theory, including monopole charges, linking structures, fragile topology, and characteristic classes [2—5]. It predicts a non-Abelian generalization of topological insulators in 1D, and it persists in the stable limit of many bands. This discovery leads to a plethora of open questions, including the search for clear experimental observables, which would be of major importance in near future.

The recipe to obtain such more general topological structures is remarkably simply: consider a partitioning of bands which differs from the most typical twofold separation into the “occupied” vs. the “unoccupied” ones. Mathematically, this corresponds to describing the band structure by flag bundles rather than by vector bundles [6]. This generalization might thus also be stimulating for research in mathematical physics. The generalized partitioning of bands is physically well motivated when describing band degeneracies, but it also arises very naturally in non-Hermitian setting [7], where the non-Abelian phenomenon results in a non-trivial interplay of invariants defined on manifolds of different dimensions.

- Q.S. Wu, A. A. Soluyanov, and
**T. Bzdušek**,*Non-Abelian band topology in noninteracting metals*, Science**365**, 1273—1277 (2019), DOI:10.1126/science.aau8740. - A. Bouhon, Q.S. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and
**T. Bzdušek**,*Non-Abelian Reciprocal Braiding of Weyl Nodes and its Manifestation in ZrTe*, Nat. Phys.**16**, 1137–1143 (2020), DOI:10.1038/s41567-020-0967-9. - A. Tiwari and
**T. Bzdušek**,*Non-Abelian topology of nodal-line rings in $\mathcal{PT}$-symmetric systems*, Phys. Rev. B**101**, 195130 (2020), DOI:10.1103/PhysRevB.101.195130. - J. Ahn, D. Kim, Y. Kim, and B.-J. Yang,
*Band Topology and Linking Structure of Nodal Line Semimetals with $\mathbb{Z}_2$ Monopole Charge*, Phys. Rev. Lett.**121**, 106403 (2018), DOI:10.1103/PhysRevLett.121.106403. - J. Ahn, S. Park, D. Kim, Y. Kim, and B.-J. Yang,
*Stiefel-Whitney classes of topological phases in band theory*, Chin. Phys. B**28**, 117101 (2019), DOI:10.1088/1674-1056/ab4d3b. - A. Bouhon,
**T. Bzdušek**, and R.-J. Slager,*Geometric approach to fragile topology beyond symmetry indicators*, Phys. Phys. B**102**, 115135 (2020), DOI:10.1103/PhysRevB.102.115135. - X.-Q. Sun, C. C. Wojcik, S. Fan, and
**T. Bzdušek**,*Alice string in non-Hermitian systems*, Phys. Rev. Research**2**, 023226 (2020), DOI:10.1103/PhysRevResearch.2.023226.

A few years back, the broad community studying topological band theory started reporting a plethora of stable band degeneracies, so-called “nodes”. Dirac points were followed by Weyl points, ushering a way towards composite fermions, nodal lines, nodal chains, and even nodal surfaces. The range of systems that can exhibit these features is also very wide, extending to superconductors, phonons, and artificially designed photonic and cold-atoms systems. The motivation to investigate these band degeneracies has been the hope that their occurrence would facilitate unusual transport phenomena or surface signatures.

I attempted to bring some order into this zoo of possibilities. By treating band nodes as singularities for spectral flattening of the Bloch Hamiltonian, I applied homotopy theory to systematically classify a wide range of band nodes and of their topological charges [1]. The strategy is, in spirit, analogous to the description of topological defects in ordered media as famously reviewed by Mermin [2]. My analysis led to the prediction of nodal surfaces in centrosymmetric time-reversal breaking superconductors such as $\textrm{SrAsPt}$ (independently predicted for $\ce{URu2Si2}$ [3]) and to the characterization of nodal rings with a monopole charge (found e.g. in certain antiferromagnets [4]). In the latter case, I implicitly stumbled upon the fragile topology, which has recently been implicated to play a role in twisted bilayer graphene near the “magic angle” [5].

The homotopic perspective on topological band theory is a gift that keeps on giving, and it allowed us to predict several phenomena which were entirely overlooked by previous studies. On one hand, in analogy with the description of surface defects (so-called “boojums”) in ordered media, we used relative homotopy to predict conversions of Weyl nodes to nodal rings [6], which we recently reported to occur at the Fermi level of $\textrm{ZrTe}$ under strain. On the other hand, by considering the action of the fundamental group on higher homotopy groups (which prominently arises in nematic liquids), we predicted an intrinsic reduction of the $K$-theoretic classification of energy bands in various systems, including dissipative photonics [7].

**T. Bzdušek**and M. Sigrist,*Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems*, Phys. Rev. B**96**, 155105 (2017), DOI:10.1103/PhysRevB.96.155105.- N. D. Mermin,
*The topological theory of defects in ordered media*, Rev. Mod. Phys.**51**, 591 (1979), DOI:10.1103/RevModPhys.51.591. - D. F. Agterberg, P. M. R. Brydon, and C. Timm,
*Bogoliubov-Fermi Surfaces in Superconductors with Broken Time-Reversal Symmetry*, Phys. Rev. Lett.**118**, 127001 (2017), DOI:10.1103/PhysRevLett.118.127001. - K. Li, C. Li, J. Hu, Y. Li, and C. Fang,
*Dirac and Nodal Line Magnons in Three-Dimensional Antiferromagnets*, Phys. Rev. Lett.**119**, 247202 (2017), DOI:10.1103/PhysRevLett.119.247202. - J. Ahn, S. Park, B.-J. Yang,
*Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle*, Phys. Rev. X**9**, 021013 (2019), DOI:10.1103/PhysRevX.9.021013. - X.-Q. Sun, S.-C. Zhang, and
**T. Bzdušek**,*Conversion Rules for Weyl Points and Nodal Lines in Topological Media*, Phys. Phys. Lett.**121**, 106402 (2018), DOI:10.1103/PhysRevLett.121.106402. - C. C. Wojcik, X.-Q. Sun,
**T. Bzdušek**, and S. Fan,*Topological Classification of Non-Hermitian Hamiltonians*, Phys. Rev. B**101**, 205417 (2020), DOI:10.1103/PhysRevB.101.205417.

In my early works, I investigated the effect of space group symmetry on the connectivity of electron energy bands. Although the role of non-symmorphic symmetry in realizing non-trivial connectivity of bands has been emphasized already two decades ago [1], its relevance for three-dimensional Dirac semimetals has been appreciated only relatively recently [2]. In work [3], we considered coupling of electron and elastic degrees of freedom in such non-symmorphic Dirac semimetals, and we showed that it may result in non-magnetic Weyl semimetals, realized through a spontaneous lattice instability. This work was developed at a time were material candidates for Weyl semimetals were still absent.

Continued considerations along these lines have eventually resulted in my most cited work to date, published in journal Nature [4]. Here, we showed that the combined presence of time-reversal symmetry and glide reflection symmetry in strongly spin-orbit coupled materials guarantees the appearance of band degeneracy along loops inside the Brillouin zone. We predicted that these loops should (1) produce a non-dispersive surface band with a very large density of states at certain energy, (2) generate anisotropic chiral anomaly for low-energy effective field theory, and (3) automatically appear near Fermi energy when a simple counting condition on the number of electrons per unit cell is fulfilled.

Even more intriguingly, we showed that a pair of glide reflections results in a pair of intersecting nodal rings, which resemble an infinite chain when viewed inside the extended momentum space. We dubbed this feature a nodal chain. A collaboration with the computational team at ETH Zürich resulted in a concrete material prediction. Nodal chains have meanwhile been observed in real materials [5] and emulated in photonic systems [6]. More recently, analysis of band connectivity inside the momentum space, and its interplay with Wyckoff positions of electron orbitals in the real space, set foundation of so-called Topological Quantum Chemistry [7] – the most exhaustive classification of topological crystalline insulators to date.

- L. Michel and J. Zak,
*Connectivity of energy bands in crystals*, Phys. Rev. B**59**, 5998 (1999), DOI:10.1103/PhysRevB.59.5998. - S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe,
*Dirac Semimetal in Three Dimensions*, Phys. Rev. Lett.**108**, 140405 (2012), DOI:10.1103/PhysRevLett.108.140405. **T. Bzdušek**, A. Rüegg, and M. Sigrist,*Weyl semimetal from spontaneous inversion symmetry breaking in pyrochlore oxides*, Phys. Rev. B**91**, 165105 (2015), DOI:10.1103/PhysRevB.91.165105.**T. Bzdušek**, Q.S. Wu, A. Rüegg, M. Sigrist, and A. A. Soluyanov*Nodal-chain metals*, Nature**538**, 75—78 (2016), DOI:10.1038/nature19099.- C.-J. Yi, B. Q. Lv, Q. S. Wu, B.-B. Fu, X. Gao, M. Yang, X.-L. Peng, M. Li, Y.-B. Huang, P. Richard, M. Shi, O. V. Yazyev, Y.-G. Shi, T. Qian, H. Ding,
*Observation of a nodal chain with Dirac surface states in $\ce{TiB2}$*, Phys. Rev B**97**, 201107(R) (2018), DOI:10.1103/PhysRevB.97.201107. - Q. Yan, R. Liu, Z. Yan, B. Liu, H. Chen, Z. Wang, L. Lu,
*Experimental discovery of nodal chains*, Nat. Phys.**14**, 461 (2018), DOI:10.1038/s41567-017-0041-4. - B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig,
*Topological quantum chemistry*, Nature**547**, 298—305 (2017), DOI:10.1038/nature23268.