Correlated Topological Insulators with Mixed Valence

We propose the local density $\mathrm{\text{approximation}}+\mathrm{\text{Gutzwiller}}$ method incorporating a Green’s function scheme to study the topological physics of correlated materials from the first principles. Applying this method to typical mixed valence materials ${\mathrm{SmB}}_6$, we find its nontrivial $Z_2$ topology, indicating that ${\mathrm{SmB}}_6$ is a strongly correlated topological insulator. The unique feature of this compound is that its surface states contain three Dirac cones in contrast to most known topological insulators.

Figure 1

(a) The CsCl-type structure of ${\mathrm{SmB}}_6$ with $Pm3m$ space group. Sm ions and ${\mathrm{B}}_6$ octahedron are located at the corner and center of the cubic lattice respectively. (b) The bulk and surface BZ for ${\mathrm{SmB}}_6$.

Figure 2

The calculated band structure for ${\mathrm{SmB}}_6$ in different energy scales by LDA [(a),(b)] and $\mathrm{LDA}+\mathrm{\text{Gutzwiller}}$ [(c),(d)]. The (b) and (d) are just the zoom in of (a) and (b) respectively, around the fermi level. Compared to the LDA results, the $4f$ $j=7/2$ bands in $\mathrm{LDA}+\mathrm{\text{Gutzwiller}}$ have been pushed up to be about 4.0 eV above the fermi level, leaving the band structure near the fermi level being dominated by $4f$ $j=5/2$ and $5d$ states. The blue, red and grey colors represent the weight of the orbital character for $5d$, $4f$, and $2p$, respectively.

Figure 3

Band hybridizations between $4f$ and $5d$ along $\Gamma$ to $X$ direction. The original $4f$ $j=5/2$ orbitals split into two ${\Gamma }_7^f$ and one ${\Gamma }_6^f$ bands. The ${\Gamma }_7^f$ bands cross and hybridize with another ${\Gamma }_7^d$ band formed by $5d$ orbitals.

Figure 4

The probability of atomic eigenstates in the ground state obtained by (a) LDA alone and (b) $\mathrm{LDA}+\mathrm{\text{Gutzwiller}}$. $N_f$ is the total number of $f$ electrons for the corresponding atomic eigenstates. The horizontal axis denotes the corresponding atomic eigenenergy.

Figure 5

The surface states of ${\mathrm{SmB}}_6$ on the (001) surface. The surface states are obtained by $\mathrm{LDA}+\mathrm{\text{Gutzwiller}}$ with a 40 layer slab on the basis of projected Wannier functions. (Inset) The fermi surfaces for surface states on (001) surface.