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

We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this $\mathbf{K}\text{−}\mathbf{b}\text{−}\mathbf{t}$ rule, the lattice topology, represented by dislocation lines oriented in direction $\mathbf{t}$ with Burgers vector $\mathbf{b}$, combines with the electronic-band topology, characterized by the band-inversion momentum ${\mathbf{K}}_{\mathrm{inv}}$, to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux $\Phi ={\mathbf{K}}_{\mathrm{inv}}·\mathbf{b}\left(\mathrm{mod}2\pi \right)$. 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 $\mathbf{K}\text{−}\mathbf{b}\text{−}\mathbf{t}$ 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 ${\mathrm{BaBiO}}_3$.

Figure 1

Illustration of the $\mathbf{K}\text{−}\mathbf{b}\text{−}\mathbf{t}$ rule relating the electronic topology in the momentum space, and the effect of dislocations in real space. (a)–(d) show the electronic-band topology of the $R$ $\mathrm{[}T\text{−}p3{\left(4\right)}_R\text{]}$ and $M$ $\mathrm{[}T\text{−}p3{\left(4\right)}_M\text{]}$ phases on a cubic lattice and the $M\text{−}R$ $\left(p{4}_{M,R}\right)$ and $X^{\prime }\text{−}R$ $\left(p{4}_{X^{\prime },R}\right)$ weak phases on tetragonal lattices. A dislocation with $\mathbf{b}={\mathbf{e}}_x$ acts on the encircled TRI momenta in the planes orthogonal to the dislocation line. As a result, the colored planes host a $\pi$ flux. The resulting number of Kramers pairs of helical modes along the edge and screw parts of the loop is indicated with a number (blue). (a) The symmetric $R$ phase has a topologically nontrivial plane hosting a $\pi$ flux orthogonal to any of the three crystallographic directions and hence any dislocation loop binds modes along the entire core, as shown for a loop in the $\widehat{x}\text{−}\widehat{z}$ plane (e). (b) In the $M$ phase, translationally active phases in the TRI planes orthogonal to $k_z$ and a crystalline phase in $k_x=\pi$ plane host $\pi$ fluxes. Hence the dislocation loop binds two pairs of modes, symmetry-protected against mixing (f). (c) In the $M\text{−}R$ phase, only the TRI planes normal to $k_z$ host a $\pi$ flux and hence the same dislocation loop binds modes only to the edge-dislocation parts (g). These modes are not protected against mixing. (d) In the $X^{\prime }\text{−}R$ phase all TRI planes orthogonal to the dislocations lines have a trivial flux, and, according to the $\mathbf{K}\text{−}\mathbf{b}\text{−}\mathbf{t}$ rule, neither the edge nor the screw dislocation of the loop binds modes (h).

Figure 2

The effect of dislocation lines in 3D TBIs. Edge and screw dislocations with $\mathbf{b}={\mathbf{e}}_x$ are treated simultaneously by considering dislocation loops of $8\times 8$ sites within the tight-binding model (3) on the lattice with $16\times 16\times 16$ sites and periodic boundary conditions [23]. (a) and (c) show the spectrum of the dislocation modes traversing the gap in case of the cubic $R$ phase $\mathrm{[}T\text{−}p3{\left(4\right)}_R\text{]}$ and tetragonal $M\text{−}R$ $\left(p{4}_{M,R}\right)$ phases, with the corresponding electronic topological configurations shown in the insets on the upper left. The circles indicate the TRI momenta hosting band inversions that yield effective $\pi$ fluxes. Additionally, the spectral density as function of the momentum $\mathbf{k}$ along the dislocation line is displayed in the insets on the lower right. We find a single Dirac cone for the $R$ phase, and a double cone for the $M\text{−}R$ phase, consistent with the $\mathbf{K}\text{−}\mathbf{b}\text{−}\mathbf{t}$ rule. The energy levels are now eightfold degenerate as both $k_z=0$ and $k_z=\pi$ planes are topologically nontrivial, and thus each yields propagating modes due to the $\pi$ flux introduced by the dislocations. The zero modes are offset due to a finite system size (see also Fig. S11 in Ref. [23]). (b) and (d) display the real space localization of the modes, where the weight of the wave function is indicated by the size of the circles and the color coding indicates the corresponding phase. Most importantly, the cubic $R$ phase features topologically protected dislocation modes propagating along the complete loop. In contrast, the $M\text{−}R$ phase binds modes only along the edge-dislocation parts. In particular, we find eight energy levels in correspondence with the number of allowed momenta along the dislocation.