In-gap collective mode spectrum of the topological Kondo insulator ${\mathrm{SmB}}_6$

Samarium hexaboride $\left({\mathrm{SmB}}_6\right)$ is the first strongly correlated material with a recognized nontrivial band-structure topology. Its electron correlations are seen by inelastic neutron scattering as a coherent collective excitation at the energy of 14 meV. Here, we calculate the spectrum of this mode using a perturbative slave-boson method. Our starting point is the recently constructed Anderson model that properly captures the band-structure topology of ${\mathrm{SmB}}_6$. Most self-consistent renormalization effects are captured by a few phenomenological parameters whose values are fitted to match the calculated and experimentally measured mode spectrum in the first Brillouin zone. A simple band-structure of low-energy quasiparticles in ${\mathrm{SmB}}_6$ is also modeled through this fitting procedure because the important renormalization effects due to Coulomb interactions are hard to calculate by ab initio methods. Despite involving uncontrolled approximations, the slave-boson calculation is capable of producing a fairly good quantitative match of the energy spectrum, and a qualitative match of the spectral weight throughout the first Brillouin zone. We find that the “fitted” band structure required for this match indeed puts ${\mathrm{SmB}}_6$ in the class of strong topological insulators. Our analysis thus provides a detailed physical picture of how the ${\mathrm{SmB}}_6$ band topology arises from strong electron interactions, and paints the collective mode as magnetically active exciton.

Figure 1

(a) A schematic band structure of ${\mathrm{SmB}}_6$. (b) Hybridization does not produce a band gap without inverted band dispersions. Note that the formation of an insulating state upon hybridization, as in (a), forms extrema in the region of the Fermi level. These extrema define a gapped “pseudo-Fermi surface” and play the prominent role in the low-energy dynamics of the spin exciton.

Figure 2

Upper left, box denotes the first Brillouin zone, while the wedge is the smallest symmetric portion of momentum space and contains all high-symmetry points. The plane containing $\Gamma ,X$, and $R$ is duplicated for comparison to Fig. 7. Lower right, ${\mathrm{SmB}}_6$ crystal structure and hopping interactions. $t_1,t_2,t_3$ are first-, second-, and third-neighbor hopping in the tight-binding model.

Figure 3

The Feynman diagram representations of the interactions between electrons mediated by slave bosons. (a) The dominant hybridization vertex. (b) A leftover from the $f$ electron hopping.

Figure 4

The ladder $\tilde{\Gamma }\left(q\right)$ and chain ${\tilde{\Gamma }}^{\prime }\left(q\right)$ diagrams that lead to the renormalized susceptibility $\chi \left(q\right)$. The wiggly lines represent the slave-boson propagators $D\left(q\right)$, while the fermion bubbles represent $\Pi \left(q\right)$.

Figure 5

The labels of vertices and propagators in bubble diagrams and their vertex corrections.

Figure 6

Energy-integrated intensity of high-symmetry planes diagrammed in Fig. 2. Left, experimental result integrated from $12–16\text{meV}$. Right, mirrored calculation from the Lindhardt function (27) integrated from $10–20\text{meV}$. The bare spectrum is inherently incoherent and requires a greater integration range. The intensity of the calculation is modulated by the $5d$ form factor to phenomenologically account for the unknown microscopic form factor.

Figure 7

Energy-resolved neutron scattering intensity along high-symmetry directions of the Brillouin zone. Dashed line is the perturbative slave-boson dispersion.