Surface-State Spin Textures and Mirror Chern Numbers in Topological Kondo Insulators

The recent discovery of topological Kondo insulators has triggered renewed interest in the well-known Kondo insulator samarium hexaboride, which is hypothesized to belong to this family. In this Letter, we study the spin texture of the topologically protected surface states in such a topological Kondo insulator. In particular, we derive close relationships between (i) the form of the hybridization matrix at certain high-symmetry points, (ii) the mirror Chern numbers of the system, and (iii) the observable spin texture of the topological surface states. In this way, a robust classification of topological Kondo insulators and their surface-state spin texture is achieved. We underpin our findings with numerical calculations of several simplified and realistic models for systems like samarium hexaboride.

Figure 1

(a) Sketch of the Dirac cones in the (001) surface Brillouin zone. (b) Positive directions of mirror invariant lines in the SBZ for the (001) surface with outward pointing normal vector ${\mathit{n}}_{\mathrm{sf}}={\mathit{e}}_z$, see Ref. [33] for further details. (c),(d) Sketch of the spin (or pseudospin, see Sec. 2) textures in the (001) SBZ. While at $\overline{\mathrm{\Gamma }}$ the winding number is always $w_{\overline{\mathrm{\Gamma }}}=1$, at the $\overline{X}$ points it can be $w_{\overline{X}}=+1$ (c) or $w_{\overline{X}}=-1$ (d), depending on the configuration of the MCNs.

Figure 2

Chiral surface states with mirror eigenvalues $\pm i$ along the $\overline{\mathrm{\Gamma }}\overline{X}$ (a) and $\overline{X}\overline{M}$ line (b) in positive direction (see Fig. 1) for $C_0=-2$ and $C_{\pi }=1$.

Figure 3

Band structure without hybridization (a) and phase diagram (b) for the ${\mathrm{\Gamma }}_8$ model defined in Eq. (9) with $t_d^{(1)}=1$, $t_d^{(2)}=-0.2$, $t_8^{(1)}=-0.03$, $t_8^{(2)}=0.02$, and ${\epsilon }_8=-3$. The two spin textures [(c),(d)] in phases I and II, respectively, are realized for the hybridization parameters $(V_8^{(1)},V_8^{(2)})=(0.3,0.07)$ and $(V_8^{(1)},V_8^{(2)})=(-0.1,0.1)$, respectively.