Potential antiferromagnetic Weyl nodal line state in ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ material

Magnetic topological semimetals have brought new insights into topological aspects because they nicely combine the band topology with intrinsic magnetic order. Comparing with ferromagnetic topological semimetals, antiferromagnetic (AFM) ones are even more special since they can break the time-reversal symmetry but show no net magnetic moment, and are highly expected for applications of topological AFM spintronics. Here, we report ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ compound is an ideal AFM Weyl nodal line semimetal. It shows time-reversal breaking Weyl nodal lines in both the spin-up and spin-down channels. Each nodal line arises from the band in one single spin channel, thus is spin-polarized. The nodal lines locate quite near the Fermi level and do not coexist with other extraneous bands. The drumhead surface state of the nodal line can be clearly identified. The nodal lines are protected by the glide mirror symmetry and are robust against spin-orbit coupling. We also show that ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ can transform into an AFM Weyl semimetal under an in-plane external magnetic field. Our results suggest ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ can serve as an excellent platform to investigate AFM topological semimetals with attractive features.

Figure 1

(a) The crystal structure of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ compound. The green, blue and red spheres denote the Li, Ti, and O atoms, respectively. (b) The ground AFM ordering in the ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ lattice. Here, only the Ti atoms are shown in the figure. The arrows in (b) denote the directions of spin in Ti atoms. (c) The primitive cell of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$. (d) Bulk and the (001) surface Brillouin zones of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$.

Figure 2

(a) The total density of states of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ under nonmagnetic and magnetic. (b) Fixed spin moment total energy as a function of constrained spin magnetization for ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ per primitive cell. Note the instability of the nonspin polarized state compared to the ferromagnetic.

Figure 3

Electronic band structure and the projected density of states of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ in (a) spin-up and (b) spin-down channels. In (a), the band crossing points in the $k$ paths $W\text{−}\mathrm{\Gamma }$ and $K\text{−}\mathrm{\Gamma }$ are denoted as P1 and P2, respectively.

Figure 4

(a) The BZ of the $k_z=0$ plane with high-symmetry $k$ points indicated. (b) Illustration of the nodal line in the $k_z=0$ plane. (c) Spin resolved band structure in high-symmetry $k$ paths in the $k_z=0$ plane. The band crossing points are denoted as P1, P2, Q1, and Q2, respectively. The bands in the spin-up and spin-down channels are shown by the blue and red lines. (d) and (e) show the shapes of nodal line from DFT calculation in the spin-up and spin-down channels. The color map indicates the local gap between two crossing bands.

Figure 5

(a) The comparison of the spin-up band structure from the Wannier model and DFT. (b) The Berry phase along the $k$ paths $W\text{−}\mathrm{\Gamma }\text{−}K$ in the $k_z=0$ plane. (c) Calculated (001) surface band structure of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ in the spin-up channel. In (c), the arrows point the drumhead surface state.

Figure 6

(a) Band structure of ${\mathrm{LiTi}}_2{\mathrm{O}}_4$ under SOC with magnetization along the [001] direction. (b) The enlarged band structures along the $\mathrm{\Gamma }\text{−}K$ and $\mathrm{\Gamma }\text{−}W$ paths under the [001] magnetization. The figure on the right panel shows the presence of nodal line (the blue ring) in this case. (c) The band structures along the $\mathrm{\Gamma }\text{−}K$ and $\mathrm{\Gamma }\text{−}W$ paths under the [110] magnetization. The figure on the right panel shows the presence of Weyl nodes (the red hollow/solid circles) in this case.

Figure 7

(a) The primitive unit cell of the lattice with four sites at positions $(\frac{1}{2},0,\pm \frac{1}{4})$ and $(0,\frac{1}{2},\pm \frac{1}{4})$. The red and blue denote the spin alignment $\pm z$. (b) Shape of the two Weyl loops (the white curves). The color denotes the local band gap between conduction and valence bands at $k_z=0$ plane. (c) Calculated electronic band structure for Ham. (3), with up-spin and down-spin bands denoted by red and blue curves. The inset of (c) shows the bulk BZ for the lattice.