We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this ${ \bf K}\text{ -}{ \bf b}\text{ -}{ \bf t}$ rule, the lattice topology, represented by dislocation lines oriented in direction ${ \bf t}$ with Burgers vector ${ \bf b}$, combines with the electronic-band topology, characterized by the band-inversion momentum ${ \bf K}_{ \rm inv}$, to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux $\Phi={ \bf K}_{ \rm inv}\cdot { \bf b}\,\, ({ \rm mod\,\,2\pi})$. Although it has already been discovered by Y. Ran { \it et al.}, Nature Phys. { \bf 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 ${ \bf K}\text{ -}{ \bf b}\text{ -}{ \bf t}$ rule . Finally, we discuss possible experimentally consequential examples in which the modes are oblivious for the direction of propagation, such as the recently proposed topologically-insulating state in electron-doped BaBiO$_3$.
Categories: cond-mat.mes-hall cond-mat.str-el