Correlation induced magnetic topological phases in the mixed-valence compound ${\mathrm{SmB}}_6$

${\mathrm{SmB}}_6$ is a mixed-valence compound with flat $f$-electron bands that have a propensity to magnetism. Here, using a realistic ${\mathrm{\Gamma }}_8$ quartet model, we investigate the dynamical spin susceptibility and describe the in-gap collective mode observed in neutron scattering experiments. We show that as the Sm valence increases with pressure, the magnetic correlations enhance and ${\mathrm{SmB}}_6$ undergoes a first-order phase transition into a metallic antiferromagnetic state, whose symmetry depends on the model parameters. The magnetic orderings give rise to distinct band topologies: while the A-type order leads to an overlap between valence and conduction bands in the form of Dirac nodal lines, the G-type order has a negative indirect gap with weak ${\mathbb{Z}}_2$ indices. We also consider the spin polarized phase under a strong magnetic field, and find that it exhibits Weyl points as well as nodal lines close to the Fermi level. The magnetic phases show markedly different surface states and tunable bulk transport properties, with important implications for experiments. Our theory predicts that a magnetic order can be stabilized also by lifting the ${\mathrm{\Gamma }}_8$ cubic symmetry, thus explaining the surface magnetism reported in ${\mathrm{SmB}}_6$.

Figure 1

(a) Sketch of the Sm ionic states and phase behavior of ${\mathrm{SmB}}_6$ under pressure. (b) Lattice structure of ${\mathrm{SmB}}_6$. (c) Crystal field splitting of the $J=5/2$ multiplet into ${\mathrm{\Gamma }}_8$ quartet and ${\mathrm{\Gamma }}_7$ doublet. The ${\mathrm{\Gamma }}_8$ quartet has a two-fold orbital degeneracy, $\alpha$ and $\beta$ Kramers doublets with elongated and planar spatial shapes. (d) Low energy band structure of ${\mathrm{SmB}}_6$ showing an insulating gap at the Fermi level. The black and red colors indicate the contributions from the highly dispersive $5d$ and quasilocalized $4f\left({\mathrm{\Gamma }}_8\right)$ states, respectively.

Figure 2

(a) Dynamical magnetic susceptibility ${\chi }^{″}(\mathit{q},\omega )$ for the valence $v=2.56$ (black) and 2.67 (blue), for $\mathit{q}$ values along the high symmetry path from $X=(\pi ,0,0)$ to $R=(\pi ,\pi ,\pi )$. The particle-hole continuum (not shown) is above $\simeq 17$ meV. (b) The G-type ordered magnetic moment $M$ as a function of $v$. The inset shows the metallic band structure at $v=2.72$.

Figure 3

Band topology of ${\mathrm{SmB}}_6$ at valence $v=2.74$ for the A-type (left) and G-type (right) antiferromagnetic states. (a), (b) Magnetic structures with moments indicated by the red arrows. (c), (d) Nodal lines and Dirac points in the vicinity (green) and below (red and purple) the Fermi level. (e)–(h) Band structures of a 100-layer slab with [001] and [010] termination, respectively. The color scale represents the wave function amplitude of the ten topmost layers. The arrows mark characteristic features of the different band topologies.