Topological Crystalline Insulators in Transition Metal Oxides

Topological crystalline insulators possess electronic states protected by crystal symmetries, rather than time-reversal symmetry. We show that the transition metal oxides with heavy transition metals are able to support nontrivial band topology resulting from mirror symmetry of the lattice. As an example, we consider pyrochlore oxides of the form $A_2{M}_2{\mathrm{O}}_7$. As a function of spin-orbit coupling strength, we find two $Z_2$ topological insulator phases can be distinguished from each other by their mirror Chern numbers, indicating a different topological crystalline insulators. We also derive an effective $\mathbf{k}·\mathbf{p}$ Hamiltonian, similar to the model introduced for ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$, and discuss the effect of an on-site Hubbard interaction on the topological crystalline insulator phase using slave-rotor mean-field theory, which predicts new classes of topological quantum spin liquids.

Figure 1

Left: Pyrochlore lattice. Transition metal ions (large balls) are located on the corner shared tetrahedra, each surrounded by an oxygen cage (smaller, red balls). Rare-earth elements fill spaces between cages. Two mirror planes are also shown. Center: Brillouin zone of face centered cubic lattice with mirror plane. Capital letters denote the high symmetry points. Right: a representation of unit cell with four sites and two possible magnetic orderings. On top ordering preserve mirror symmetry and on bottom ordering is of all-in-all-out which breaks mirror symmetry.

Figure 2

Berry curvature $\Omega (\mathbf{k})$ along the boundary of the mirror plane in Brillouin zone. $L_1=(\pi ,\pi ,\pi )$ and $L_2=(\pi ,-\pi ,\pi )$ are high symmetry points of the 3D Brillouin zone residing on the boundary of the mirror plane in $k$ space. In (a)–(b) the time reversal symmetry is preserved and $\lambda <{\lambda }_c$ in (a) and $\lambda >{\lambda }_c$ in (b). In (c)–(d) the time reversal symmetry is broken by the magnetic ordering shown in Fig. 1 which preserves and breaks the mirror symmetry in (c) and (d), respectively.

Figure 3

The band structure of the tight-binding model in Eq. (1) along the high symmetry points of projected Brillouin zone as shown in inset of (a). In panels (a)–(b) the time reversal symmetry is preserved and $\lambda =3$ (a) $\lambda =4$ (b). In panels (c)–(e) the time reversal symmetry is broken, but the mirror symmetry is preserved: (c) magnetic field $\mathbf{B}=B(1,0,1)$, inset enlarges the crossing (d) magnetic field $\mathbf{B}=B(-1,0,1)$ and (e) magnetic ordering. In panel (f) both time reversal and mirror symmetries are broken by a magnetic ordering of all-in-all-out type. Green and red lines correspond to surface sates on bottom and top surfaces, respectively, of slab. Dark blue lines are bulk sates.