Weyl-Kondo semimetals in nonsymmorphic systems

There is considerable current interest to explore electronic topology in strongly correlated metals, with heavy fermion systems providing a promising setting. Recently, a Weyl-Kondo semimetal phase has been concurrently discovered in theoretical and experimental studies. The theoretical work was carried out in a Kondo lattice model that is time-reversal invariant but inversion-symmetry breaking. In this paper, we show in some detail how nonsymmorphic space-group symmetry and strong correlations cooperate to form Weyl nodal excitations with highly reduced velocity and pin the resulting Weyl nodes to the Fermi energy. A tilted variation of the Weyl-Kondo solution is further analyzed here, following the recent consideration of such effect in the context of understanding a large spontaneous Hall effect in ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ (Dzsaber et al., arXiv:1811.02819). We discuss the implications of our results for the enrichment of the global phase diagram of heavy fermion metals, and for the space-group symmetry enforcement of topological semimetals in other strongly correlated settings.

Figure 1

(a) Quantum phase transition of heavy fermion metals. The vertical axis varies temperature $T$ and thus the amount of thermal fluctuations. The horizontal axis tunes a nonthermal control parameter $\delta$, which corresponds to the ratio of the Kondo to RKKY interactions. The red point labeled ${\delta }_{\text{QCP}}$ marks a quantum critical point where the ordered antiferromagnetic phase and the disordered heavy Landau-Fermi liquid phases meet at $T=0$, producing the quantum critical non-Fermi-liquid regime at nonzero temperatures. (b) Heavy fermion global phase diagram [6, 7, 8, 9, 10, 11]. The parameter $G$ controls the amount of geometric frustration of local moments, while $J_K$ modifies the magnitude of the Kondo coupling. AF labels phases with antiferromagnetic ordering of local moments, and P labels paramagnetic phases. Subscript $L$ denotes large Fermi-surface phases, and subscript $S$ denotes small Fermi-surface phases. The dashed lines, labeled “I,” “II,” and “III,” illustrate three trajectories [corresponding to three cuts in the parameter space for the tuning parameter $\delta$ shown in Fig. 1] of quantum phase transitions. Near the border of the AF phase boundaries, large magnetic fluctuations may lead to emergent correlated topological phases in heavy fermion systems when the SOC is large.

Figure 2

Eigenenergy of the surface states of the [001] plane, with parameters $(E_d,\ell ,r,V)=(-7,7.28,0.22,7.5)$. Blue and red points show the position of the Weyl and anti-Weyl nodes of the Brillouin-zone boundary, and thick black lines show the Fermi arcs connecting nodes to their opposite chirality partner on the four neighboring Brillouin-zone boundaries.

Figure 3

Model in the absence of the Kondo effect, $V=0$. Dispersion along a high-symmetry path in the fcc Brillouin zone. The localized electrons' energy level is at the Fermi energy $E_F$. (a) Fourfold-degenerate line nodes along $X-W$ when SOC $\lambda =0$ and inversion symmetry is preserved, $m=0$. (b) Dirac node develops at $X$ from (a) when SOC is present, $\lambda >0$. (c) Weyl nodes emerge from (b) when inversion symmetry is broken and $0<\frac{m}{4\lambda }<1$. $D\sim 8t$ denotes the unrenormalized conduction electron bandwidth.

Figure 4

Kondo effect-driven Weyl nodes. Dispersion along a high-symmetry path in the fcc Brillouin zone. Weyl-Kondo semimetal with $V=7.5$ and nodes pinned at $E_F$. This develops from Fig. 3 when the strong coupling is renormalized, $\tilde{V}\to rV>0$. The bandwidth of the upper quartet of bands is approximately that of the conduction electron bandwidth $\sim D$, whereas the strongly renormalized lower quartet of heavy bands corresponds to the $k_B{T}_K$ energy scale. The parameters are $(E_d,\ell ,r,V,\lambda ,m)\simeq (-7,7.279,0.220,7.5,0.5,1)$.

Figure 5

Projected density of states, showing full energy range corresponding to Fig. 4. Inset: zoomed-in to localized set of bands near the nodes at $E_F$. Shades of red indicate contributions from $f$ fermions, and shades of blue indicate contributions from $c$ electrons. The parameters are $(E_d,\ell ,r,V,\lambda ,m)\simeq (-7,7.279,0.220,7.5,0.5,1)$.

Figure 6

High-symmetry dispersion of the tilted WKSM model. (a) High-symmetry contour in green along the $\mathbf{k}=(k_x,k_y,2\pi )$ Brillouin-zone boundary plane, through (anti)nodes marked in (red) blue. The square marks the small region of the Brillouin zone over which the Berry-curvature distribution is shown in Fig. 7. (b) Bulk dispersion along the contour of the tilted model with $C=0.8$, other parameters are found self-consistently to be $(E_d,\ell ,r,V,\lambda ,m)\simeq (-7,7.282,0.222,7.5,0.5,1)$. (c) Bulk dispersion with $C=0$, and the same parameters Fig. 4.

Figure 7

Berry curvature strength captured by Fermi surfaces. The Berry curvature component ${\mathrm{\Omega }}_{yz}$ [Eq. (21)] of the WKSM phase in the [001] plane of the Weyl node located at the center of the plot, $\mathbf{k}=(\frac{\pi }{3},0,2\pi )$ (black dot), which corresponds to the right red point of Fig. 6, and Fermi surfaces from a slightly metallic filling $n_d+n_c=1+{10}^{-5}$, sharing the parameters $(E_d,V,\lambda ,m)\simeq (-7,7.5,0.5,1)$. (a) The solid contour is the intersection of the 3D Fermi surface of the model with the BZ-boundary plane (i.e., for $k_z=2\pi$) without tilting, and self-consistently determined parameters $(C,\mu ,r,\ell )\simeq (0,-9.748,0.220,7.279)$. The dashed contour is the counterpart for the Fermi surface of the tilted model, with parameters $(C,\mu ,r,\ell )\simeq (0.9,-9.811,0.222,7.281)$. (b) Same as (a), but with an even smaller plot range in $\mathbf{k}$ space. The full range of the Berry curvature is truncated, and the deep blue and red regions near the node represent values that extend beyond the legend. Note that, in the 3D BZ, the Fermi pockets in the two cases have the same volume.