Cubic Topological Kondo Insulators

Current theories of Kondo insulators employ the interaction of conduction electrons with localized Kramers doublets originating from a tetragonal crystalline environment, yet all Kondo insulators are cubic. Here we develop a theory of cubic topological Kondo insulators involving the interaction of ${\Gamma }_8$ spin quartets with a conduction sea. The spin quartets greatly increase the potential for strong topological insulators, entirely eliminating the weak topological phases from the diagram. We show that the relevant topological behavior in cubic Kondo insulators can only reside at the lower symmetry $X$ or $M$ points in the Brillouin zone, leading to three Dirac cones with heavy quasiparticles.

Figure 1

Contrasting the phase diagram of (a) tetragonal [27] and (b) cubic topological Kondo insulators. Cubic symmetry extends the strong topological insulator phase into the Kondo limit. For ${\mathrm{SmB}}_6$ $v=3-n_f$ gives the valence of the Sm ion, while $n_f$ measures the number of $f$ holes in the filled $4f^6$ state, so that $n_f=1$ corresponds to the $4f^5$ configuration.

Figure 2

Schematic band structure illustrating (a) one-dimensional Kondo insulator with local cubic symmetry and (b) three-dimensional cubic Kondo insulator. Hybridization between a quartet of $d$ bands with a quartet of $f$ bands leads to a Kondo insulator. The fourfold degeneracy of the $f$ and $d$ bands at the high symmetry $\Gamma$ and $R$ points of the Brillouin zone guaranties that the three-dimensional topological invariant is determined by the band inversions at the $X$ and $M$ points only.

Figure 3

Temperature dependence of the hybridization gap parameter $b$ and the renormalized $f$-level position (inset) for various values of the bare hybridization (see Supplemental Material [28] for more details).

Figure 4

Band structure consistent with PES and LDA studies of ${\mathrm{SmB}}_6$ computed with the following parameters: $n_f=0.48$ (or $b=0.73$), $V=0.4\mathrm{eV}$,$t_d=2\mathrm{eV}$, ${\mu }_d=0.2\mathrm{eV}$, $\eta ={\eta }^{\prime }=-.3$, ${ϵ}_f=-0.01\mathrm{eV}$ (${ϵ}_{f0}=-0.17\mathrm{eV}$), $t_f=-.05\mathrm{eV}$, $T={10}^{-4}\mathrm{eV}$, and the gap is $\Delta =12\mathrm{meV}$. Shaded region denotes filled bands. Inset shows the ground-state energy computed for a slab of 80 layers to illustrate the three gapless surface Dirac excitations at the symmetry points $\widehat{\Gamma }$, ${\widehat{X}}^{\prime }$, and $X^{\prime \prime }$.