Tunable topological semimetallic phases in Kondo lattice systems

We exploit topological semimetallic phases resulting from the Kondo screening in Anderson lattice models. It is shown that by including spin-orbit interactions both in the bulk electrons and in the hybridization between the conduction electrons and electrons in the $f$ orbit, all types of topological semimetallic phases can be realized in Anderson lattice models. Specifically, upon either broken time-reversal symmetry or broken inversion symmetry, we find that either the Weyl semimetallic phase, Dirac semimetallic phase, or nodal-ring semimetallic phases always emerge between insulating phases and can be accessed by tuning either temperature or spin-orbit interaction. For Anderson lattice models with general three-dimensional spin-orbit hybridization between the conduction electrons and electrons in the $f$ orbit, we find that Weyl nodal-ring semimetallic phases emerge between strong and weak topological insulating phases. Furthermore, in the presence of an exchange field, Weyl semimetallic phases form after two Weyl points of charge $\pm 1$ split off from a Dirac point at time-reversal momenta. On the other hand, when the spin-orbit interaction is included in the conduction electron, we find that upon the rotation symmetry being broken with anisotropic hopping amplitudes, a Weyl semimetallic phase emerges with a double Weyl node with charges of $\pm 2$. Furthermore, the Weyl semimetallic phases with charges of $\pm 2$ can be tuned into Weyl semimetallic phases with charges of $\pm 1$ through the inclusion of the Rashba spin-orbit interaction. Our analyses indicate that Anderson lattices with appropriate spin-orbit interactions provide a platform for realizing all types of topological semimetallic phases.

Figure 1

(a) Phase diagram of the Anderson lattice model with an external magnetic field. Here ${\varepsilon }_{\lambda }\equiv ({\varepsilon }_d+\lambda )/(1+\eta r^2)$, ${\varepsilon }_{{\lambda }_{so}}\equiv r{\lambda }_{so}/(1+\eta r^2)$, and ${\varepsilon }_M\equiv M/(1+\eta r^2)$. The shaded regime is the Weyl semimetallic phase, while white regimes are insulating phases with gaps in electronic structures. When ${\varepsilon }_M=0$, the gapless phases at ${\varepsilon }_{\lambda }/t=-6,-2,2,6$ are Dirac semimetallic phases with corresponding Dirac points being at time-reversal momenta $\mathrm{\Gamma },X,M,R$, respectively. (b) Emergence of a finite-temperature Weyl point that splits off from the $M$ point. Here $t=1$, ${\lambda }_{so}=0.14$, $\eta =0.05$, ${\varepsilon }_d=1.837$, $M_z=0.001$. The transition temperature when the Kondo insulator becomes a Weyl semimetal occurs at $T_W=0.03$.

Figure 2

Sketch of the distribution of Weyl nodes in the first Brillouin zone. The monopole charges of each Weyl nodes are denoted by $-$ or $+$. Weyl nodes move along the dashed lines when the temperature changes. Notice that in order to display Weyl nodes clearly, here $E\left(k\right)$ does not include the effective chemical potential ${\mu }_{\mathbf{k}}$.

Figure 3

(a) Topological phase diagram in the presence of constant hybridization $v_0$ between $c$ and $f$ electrons; shaded regimes indicate gapless phases, while white regimes are insulating phases, labeled by a strong topological insulator (STI), weak topological insulator (WTI), and Kondo insulator (KI) when the valence bands are filled [20]. Here ${\varepsilon }_{\lambda }\equiv \frac{{\varepsilon }_d+\lambda }{1+\eta r^2}$ and $({\nu }_0;{\nu }_1,{\nu }_2,{\nu }_3)$ are topological indices. (b) The phase transition path (the red dashed line) in the phase diagram with $t=1$, ${\varepsilon }_d=1.724$, $v_0=0.01$, ${\lambda }_{so}=0.2$. When the temperature increases from $T=0.02$ to $T=0.04$, the Anderson lattice goes through Weyl nodal-ring semimetallic phases.

Figure 4

Topological phase diagram in the presence of the bulk spin-orbit interaction in $c$ and $d$ electrons with equal strength. Here the hopping amplitude is isotropic, with $a_z=a_y=a_x=1$. The shaded regime is the Weyl semimetallic phases, while the white regime is the Kondo insulating phase.

Figure 5

(a)Topological phase diagram when the inversion symmetry of the Anderson lattice is broken. Here the $z$ axis is the high-symmetry axis, and the anisotropy of the hopping amplitude is characterized by $a_x=1$, $a_y=0.5$, $a_z=0.08$. Here gray, green, blue, and purple regimes are Weyl semimetallic phases with Weyl nodes emerging at ${\mathbf{k}}_{\text{w}}=(0,0,k_{{\text{w}}_z})$, $(0,\pi ,k_{{\text{w}}_z})$, $(\pi ,0,k_{{\text{w}}_z})$, and $(\pi ,\pi ,k_{{\text{w}}_z})$, respectively, while the red regime is the overlap regime with the emergence of both Weyl nodes from the overlapping Weyl semimetallic phases. (b) Emergence of the Weyl semimetallic phase at finite temperature. Here Weyl nodes emerge in the $k_z$ axis from $X(\pi ,0,0)$ to $M(\pi ,0,\pi )$. Note that two intersecting points in the inset may look like a nodal ring. A clear demonstration of these intersecting points being Weyl points is shown in Fig. 6. Parameters are $t=1$, ${\lambda }_{so}=0.2$, $\eta =0.05$, ${\varepsilon }_d=0.705$, ${\tilde{\lambda }}_{so}=0.001$, $a_y=0.5$, and $a_z=0.001$. The critical temperature for the emergence of Weyl nodes is $T_W=0.03$.

Figure 6

Sketch of the distribution of Weyl nodes in the first Brillouin zone for Kondo-Weyl semimetals without inversion symmetry. The monopole charge of each Weyl node is $+2$ or $-2$ and is denoted by $+$ or $-$, respectively. Weyl nodes will move along the dashed lines with changing parameters of the system. Here parameters are $t=1$, ${\lambda }_{so}=0.3$, $\eta =0.05$, ${\varepsilon }_d=1.305$, ${\tilde{\lambda }}_{so}=0.3$, $a_y=0.5$, $a_z=0.1$, and $T=T_W=0.03$. Notice that in order to display Weyl nodes clearly, here $E\left(k\right)$ does not include the effective chemical potential ${\mu }_{\mathbf{k}}$.

Figure 7

(a) Topological phase diagram when the hybridization is governed by Rashba interaction. Here the $z$ axis is the high-symmetry axis, and the anisotropy of the hopping amplitude is characterized by $a_x=1$, $a_y=0.5$, $a_z=0.08$. Gray, green, blue, and purple regimes are Weyl semimetallic phases with Weyl nodes emerging at ${\mathbf{k}}_{\text{w}}=(0,0,k_{{\text{w}}_z})$, $(0,\pi ,k_{{\text{w}}_z})$, $(\pi ,0,k_{{\text{w}}_z})$, and $(\pi ,\pi ,k_{{\text{w}}_z})$, respectively. (b) Emergence of the Weyl semimetallic phase at finite temperature. Here Weyl nodes emerge in the $k_z$ axis from $X(\pi ,0,0)$ to $M(\pi ,0,\pi )$. Note that two intersecting points in the inset may look like a nodal ring. A clear demonstration of these intersecting points being Weyl points is confirmed by similar plots shown in Fig. 6. Parameters are $t=1$, ${\lambda }_{Ra}=0.2$, $\eta =0.05$, ${\varepsilon }_d=0.978$, ${\tilde{\lambda }}_{so}=0.001$, $a_y=0.5$, and $a_z=0.001$. The critical temperature for emergence of Weyl nodes is $T_W=0.03$.