Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band insulators

Abstract

We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this K-b-t rule, the lattice topology, represented by dislocation lines oriented in direction t with Burgers vector b, combines with the electronic-band topology, characterized by the band-inversion momentum K inv , to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux = K inv · b(mod 2π ). Although it has already been discovered by Ran et al. [Nat. Phys. 5, 298 (2009)] that dislocation lines host propagating modes, the exact mechanism of their appearance in conjunction with the crystal symmetries of a topological state is provided by the K-b-t rule. Finally, we discuss possible experimentally consequential examples in which the modes are oblivious to the direction of propagation, such as the recently proposed topologically insulating state in electron-doped BaBiO 3 . Topological band insulators (TBIs) represent a new class of quantum materials that, due to the presence of time-reversal symmetry (TRS), feature an insulating bulk band gap together with metallic edge or surface states protected by a Z 2 topological invariant [1][2][3][4]. This Z 2 classification of TBIs is a part of the classification of free gapped fermion matter in the presence of the fundamental antiunitary time-reversal and particle-hole symmetries, the so-called tenfold way [5][6][7]. On the other hand, topologically insulating crystals break continuous translational and rotational symmetries down to discrete symmetries mathematically characterized by the space groups. By considering the crystal symmetries, an extra layer in this Z 2 classification of TBIs has been recently uncovered [8]. This space group classification of TBIs results in the enrichment of the tenfold way with extra phases, such as crystalline (or "valley") phases [9] and translationally active phases, the latter featuring an odd number of band inversions at non-points in the Brillouin zone (BZ) [8,10]. Dislocations are of central interest in this endeavor, being the topological defects exclusively related to the lattice translations. In two dimensions (2D), the role of these lattice defects has been recently elucidated in TBIs [10,11], as well as in topological superconductors [12,13] and interacting topological states [14][15][16]. In particular, it has been shown that these lattice defects in two-dimensional TBIs act as probes of distinct topological states through binding of the localized zero-energy modes [10].Although early on it was identified that in three-dimensional TBIs dislocation lines support propagating helical modes [17], the precise role of dislocations has not been explored thoroughly [18][19][20]. In particular, the relation between the lattice symmetry and the electronic topology, as well as the characterization of these topological states through the response of the dislocation lines, still needs to be addressed. Dislocations in thr...

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