Distinct Topological Crystalline Phases in Models for the Strongly Correlated Topological Insulator ${\mathrm{SmB}}_6$

${\mathrm{SmB}}_6$ was recently proposed to be both a strong topological insulator and a topological crystalline insulator. For this and related cubic topological Kondo insulators, we prove the existence of four different topological phases, distinguished by the sign of mirror Chern numbers. We characterize these phases in terms of simple observables, and we provide concrete tight-binding models for each phase. Based on theoretical and experimental results for ${\mathrm{SmB}}_6$ we conclude that it realizes the phase with ${\mathcal{C}}_{k_z=0}^{+}=+2$, ${\mathcal{C}}_{k_z=\pi }^{+}=+1$, ${\mathcal{C}}_{k_x=k_y}^{+}=-1$, and we propose a corresponding minimal model.

Figure 1

(a) 3D BZ and its mirror planes $k_z=0$, $k_z=\pi$, $k_x=k_y$; high-symmetry points are $\mathrm{\Gamma }=(0,0,0)$, $X=(\pi ,0,0),(0,\pi ,0),(0,0,\pi )$, $M=(0,\pi ,\pi ),(\pi ,0,\pi ),(\pi ,\pi ,0)$, $R=(\pi ,\pi ,\pi )$. (b) Corresponding 2D BZ for a (001) surface with its high-symmetry points $\overline{\mathrm{\Gamma }}=(0,0)$, $\overline{X}=(\pi ,0)$, ${\overline{X}}^{\prime }=(0,\pi )$ and $\overline{M}=(\pi ,\pi )$ and its mirror planes $k_{x,y}=0$, $k_{x,y}=\pi$, $k_x=\pm k_y$, which are pairwise equivalent by $C_{4v}$ symmetry. (Planes at $k_{x,y}=\pi$ and $k_{x,y}=-\pi$ are identical and share a single label.) Also shown is the 2D Fermi surface of ${\mathrm{SmB}}_6$, consisting of three pockets centered at $\overline{\mathrm{\Gamma }}$, $\overline{X}$, and ${\overline{X}}^{\prime }$. Dots mark its intersections with mirror planes.

Figure 2

Surface-state mirror eigenvalues $\pm i$ (defined at surface momenta where the Fermi contour crosses a mirror plane) for the four phases with MCNs $({\mathcal{C}}_{k_z=0}^{+},{\mathcal{C}}_{k_z=\pi }^{+},{\mathcal{C}}_{k_x=k_y}^{+})$ as quoted in panels (a–d). The $\mathcal{C}$ signs can be condensed into $v\equiv \mathrm{sgn}({\mathcal{C}}_{k_z=0}^{+}{\mathcal{C}}_{k_x=k_y}^{+})$ and $w\equiv \mathrm{sgn}({\mathcal{C}}_{k_z=0}^{+}{\mathcal{C}}_{k_z=\pi }^{+})$, see text. Green arrows show the surface-state SEV for ${\mathrm{\Gamma }}_8$ states; for ${\mathrm{\Gamma }}_7$ states (see Table 1 for models) it would be reversed. In the FTB model this reversal depends on the relative weight of ${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_8$ states, with ${\mathrm{\Gamma }}_8$ dominating for realistic parameters [24].

Figure 3

Isoenergy contours with SEV [(a),(c)] and QPI signal [(b),(d)] at the Fermi level for two different ${\mathrm{SmB}}_6$ tight-binding models. Large arrows show intercone scattering wave vectors. (a),(b): Four-band model $sf$ in the $(-2,+1,-1)$ phase ($w=-1$); (c),(d): Eight-band model ${\mathrm{\Gamma }}_8^{2z}$ in the $(+2,+1,-1)$ phase ($w=+1$); with parameters given in the Supplemental Material [25]. An intercone QPI signal is seen in (b), but not in (d), dictated by the surface-state spin structure.