Symmetry-protected spinful magnetic Weyl nodal loops and multi-Weyl nodes in $5d^n$ cubic double perovskites $(n=1,2)$

Using both an effective three-band model and ab initio calculations, we have investigated various topological features in the cubic ferromagnetic $5d^{1,2}$ systems showing large spin-orbit coupling (SOC): ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6, {\mathrm{Sr}}_2{\mathrm{SrOsO}}_6$, and ${\mathrm{Ba}}_{2}B{\mathrm{ReO}}_6$ ($B$=Mg, Zn). In the presence of time-reversal symmetry ($\mathcal{T}$), spinless Dirac nodal loops linked to each other at the $W$ points appear in the mirror planes. Remarkably, breaking $\mathcal{T}$ leads to spinful magnetic Weyl nodal loops (MWNLs) that are robust even at large SOC and correlation strength $U$ variation due to the combination of mirror symmetry and broken $\mathcal{T}$. Additionally, there are two types of magnetic Weyl points with chiral charges $\left|\chi \right|=1,2$ along the $C_{4v}$ symmetry line, and another type-II MWNL encircling the zone center, that are dependent on $U$. Furthermore, the ferromagnetic ${\mathrm{Ba}}_2{\mathrm{ZnReO}}_6$ is an ideal half semimetal with MWNLs and magnetic Weyl nodes at the Fermi level without the interference of topologically trivial bulk states. These systems give rise to a remarkably large anomalous Hall conductivity ${\sigma }_{xy}$ of up to 1160 (${\mathrm{\Omega }\mathrm{cm})}^{-1}$. Our findings may apply widely for $t_{2g}$ systems with cubic (or slightly distorted) fcc-like structures.

Figure 1

(a) Brillouin zones (BZs) of bulk and (001) surfaces with high-symmetry points of the fcc structure. (b) Spinless nodal loops, penetrating the BZ, in the $\mathcal{T}$-preserved case. The light (green) colored loops are represented in the extended zone scheme for better visualization. As shown on the front face, each loop is linked at the $W$ points at the middle of each side.

Figure 2

Evaluation of $\mathcal{T}$-preserved band structures of the three-band model, when (a) only $t_{\sigma }$, (b) both $t_{\sigma }$ and $t_{\pi }$, and (c) additional $t_{\delta }$ and SOC of ${\xi }_{\text{SOC}}=0.2$ eV were included. In these plots, $t_{\sigma }=-0.2$ eV is used for illustration. In (b) and (c) we also used $t_{\pi }=-t_{\sigma }/4$ and $t_{\delta }=t_{\pi }/2$. The arrows in (b) denote DPs. The horizontal (blue) dashed lines indicate each Fermi level $E_F$ of the $d^n$ configuration ($n=1,2,3$).

Figure 3

(a) Band structure of the three-band model, when breaking $\mathcal{T}$ by ${\mathrm{\Delta }}_{\text{ex}}=0.3$ eV in Fig. 2. Along the $\mathrm{\Gamma }-Z$ line, two types of WPs, denoted by W1 and W2, emerge. The $\pm i$ indicate the mirror eigenvalues of the two bands that produce nodal loops. The Fermi energies of $d^n$ ($n=1,2,3$) are denoted by the (blue) dashed lines. (b) Band dispersions around the W2 point on a plane perpendicular to the $k_z$ axis, implying an exotic Weyl node. (c) Plots of the Wannier charge center (WCC) of the (top) W1 and (bottom) W2 points. (d) Band-resolved chiral charge ${\chi }_n\left(k_z\right)$ along the $k_z$ axis for the lowest three bands (labeled in order of lower energy).

Figure 4

Band structures and surface spectra with densities of states (DOSs) in nonmagnetic ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ (top) and ${\mathrm{Sr}}_2{\mathrm{SrOsO}}_6$ (bottom). These contain only the isolated and partially filled $t_{2g}$ manifolds. (a, d) Both GGA (black solid) and $\mathrm{GGA}+\mathrm{SOC}$ (red dashed) band structures. The band structure of ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ is consistent with the previous one in Ref. [1]. Surface spectra and DOSs are shown without SOC in (b) and (e), and with SOC in (c) and (f). The surface DOSs are given in arbitrary units. The surface states lead to sharp peaks in the surface DOSs. In the surface spectra plots, obtained in the conventional cubic cell, the strong yellowish lines represent the surface states. (g) WCCs of ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ at the $k_z=0$ plane, indicating the mirror Chern number ${\mathcal{C}}_M=2$.

Figure 5

FM band structures containing only the $t_{2g}$ manifold of ${\mathrm{Sr}}_2{\mathrm{SrOsO}}_6$ in $\mathrm{GGA}+\mathrm{SOC}$(001)$+U$ with (a) $U_{\text{eff}}=0$ and (b) 4 eV. The latter represents the observed insulating phase. The former is very similar to the results of our model calculations given in Fig. 3. In (a), W1 and W2 indicate WPs with the chiral charges of $\left|\chi \right|=1,2$, respectively, as in Fig. 3. Inclusion of correlation leads to another nodal loop around the $\mathrm{\Gamma }$ point, shown in the inset of (b). In (b), a pair of WPs appears only at 0.7 eV along the $\mathrm{\Gamma }-Z$ line. Here, all symmetry points, except for the $W^{\prime }$ and $Z$ points, are on the $k_z=0$ plane.

Figure 6

(a) Magnetic Weyl nodal loops, appearing on the $M_z$ plane, of the insulating FM ${\mathrm{Sr}}_2{\mathrm{SrOsO}}_6$ for GGA$+$SOC(001)$+U$ with $U_{\text{eff}}=4$ eV. The BZ-penetrating loops are given only for the filled bands. (b, c) Berry phases along the $\mathrm{\Gamma }-K$ and the $\mathrm{\Gamma }-X$ lines for closed paths crossing the loops, respectively. (d) Band dispersions for the $\mathrm{\Gamma }$-centered nodal loops in the perpendicular and tangential directions to the $k_z=0$ plane in $\mathrm{GGA}+\mathrm{SOC}$(001)$+U$ at $U_{\text{eff}}=4$ eV.

Figure 7

Energy dependent anomalous Hall conductivity ${\sigma }_{xy}$ of ${\mathrm{Sr}}_2{\mathrm{SrOsO}}_6$ in $\mathrm{GGA}+\mathrm{SOC}$(001)$+U$ at $U_{\text{eff}}=4$ eV. The strong anisotropy in the SOC leads to sharp peaks at the energy levels of the MWNLs.

Figure 8

Left: $\mathrm{GGA}+\mathrm{SOC}$(001) band structure of FM ${\mathrm{Ba}}_2{\mathrm{ZnReO}}_6$, enlarged in the range of –0.5 to 0.5 eV. Two types of WPs, W1 and W2, are denoted by (violet) circles along the $\mathrm{\Gamma }-Z$ line. Other WPs at the $W$ point and along the $\mathrm{\Gamma }-K$ line lead to a nodal loop (see text). Center: The corresponding spin-polarized total DOS, indicating a half-semimetallic character. Right: Energy dependent AHC ${\sigma }_{xy}$, showing two sharp peaks at 0.15 eV and at precisely $E_F$.

Figure 9

(a) (001) surface spectrum and DOS (in arbitrary units) in $\mathrm{GGA}+\mathrm{SOC}$(001) of ${\mathrm{Ba}}_2{\mathrm{ZnReO}}_6$. The surface DOS shows a sharp peak at $E_F$ that results from the drumhead state. (b) Evaluation of surface spectra on isoenergy contours at $E_F$ –5, 0, $+5$, and $+10$ (in units of meV). The strong yellowish lines denote the surface state.