Competing exotic topological insulator phases in transition-metal oxides on the pyrochlore lattice with distortion

In this work we investigate the phase diagram of heavy ($4d$ and $5d$) transition-metal oxides on the pyrochlore lattice, such as those of the form ${\mathrm{A}}_2{\mathrm{M}}_2{\mathrm{O}}_7$, where A is a rare-earth element and M is a transition-metal element. We focus on the competition between Coulomb interaction, spin-orbit coupling, and lattice distortion when these energy scales are comparable. Strong spin-orbit coupling entangles the spin and the $t_{2g}$ $d$ orbitals giving rise to doublet $j=1/2$ and quadruplet $j=3/2$ states. In contrast to previous works which focused on the doublet manifold, we also discuss the quadruplet manifold which is relevant for several pyrochlore oxides. The Coulomb interaction is taken into account by use of the slave-rotor mean-field theory and different classes of lattice distortions which further split the levels of the quadruplet $j=3/2$ manifold are studied. Various topological phases are predicted, including exotic strong and weak topological Mott insulating phases. We discuss the general structure of the phase diagram for several values of $d$-shell filling and various symmetry classes of lattice distortions. Our results are relevant to the search for exotic topological insulators and quantum spin liquids in strongly correlated materials with strong spin-orbit coupling.

Figure 1

(a) An illustration of the pyrochlore lattice which is composed of corner sharing tetrahedra. Transition elements are indicated by black solid circles. (b) Each transition ion is surrounded by an oxygen octahedron shown by six solid blue (dark gray) circles. A transition ion is located at the origin of the local coordinate and is shown in black. We study a trigonal distortion preserving $C_3$ symmetry applied along the [111] direction (or its equivalent), shown by two yellow (gray) faces, and an elongation preserving $C_4$ symmetry along the $z$ axis of the local coordinate. (c) A schematic representation of the splitting of the bare atomic $d$ levels (1), due to a cubic crystal field arising from the octahedral environment (2), unquenched spin-orbit coupling in the $t_{2g}$ manifold (3), and a distortion of the octahedron (4). The values of the splittings in (4) depend on $\lambda$ and ${\Delta }_{3,4}$.

Figure 2

Phase diagram of the $j_{\mathrm{eff}}=1/2$ band model corresponding to $n_d=5$ with positive ${\Delta }_3=2t$ (upper panel) and negative trigonal distortion ${\Delta }_3=-2t$ (lower panel). The dashed line separates the rotor condensed phases (below) from the uncondensed phases (above). We set $t=1$, and the phases labeled are as follows: Strong topological insulator (STI), weak topological insulator (WTI), gapless Mott insulator (GMI), topological Mott insulator (TMI), weak topological Mott insulator (WTMI), and metallic phases.

Figure 3

Phase diagram of $j=3/2$ band model on the undistorted lattice for $n_d=3$. The abbreviations used are the same as those in Fig. 2. The dashed line separates the rotor condensed phase (below) from rotor uncondensed phase (above). All energies are expressed in units of $t$, as before. Compared to the corresponding phase diagram for $j=1/2$ with $n_d$ $=$ 5, the STI and TMI occupy a much smaller portion of the phase diagram (Ref. 15). The phase diagram for $j=2/3$ with $n_d=2$ has no topologically nontrivial phases within our model, even in the presence of distortion.

Figure 4

Phase diagram of the $j=3/2$ band model with $n_d=3$, including the trigonal distortion of the octahedra. The labeling of the phases is the same as that used in Fig. 2. In the upper panel ${\Delta }_3=2t$. In the lower panel the interaction is fixed at $U=4t$ and the strength ${\Delta }_3$ of the trigonal distortion is varied, illustrating possible phases that may arise upon the application of pressure to a real system. All energies are expressed in units of $t$. In both phase diagrams the dashed line separates the rotor uncondensed phase (above) from the condensed phase (below). We note the “pocket” of STI around $\lambda \approx 1,{\Delta }_3\approx 1.5$ in the lower figure has a numerically difficult to determine boundary with the metallic phase; we have presented our best assessment.

Figure 5

Phase diagram of the $j=3/2$-band model with $C_4$ tetragonal distortion. The upper and lower panel correspond to compression and elongation distortion of octahedra, respectively. We set ${\Delta }_4=2t$ for compression and ${\Delta }_4=-2t$ for elongation. All energies are expressed in units of $t$. In both phase diagrams the dashed line separates the rotor uncondensed phase (above) from the condensed phase (below). Note that for these values of distortion, the topological phases are interaction-driven, i.e. they do not extend down to the non-interacting limit as they do in Figs. 2-4.