Topological Order in a Correlated Three-Dimensional Topological Insulator

Motivated by experimental progress in the growth of heavy transition metal oxides, we theoretically study a class of lattice models of interacting fermions with strong spin-orbit coupling. Focusing on interactions of intermediate strength, we derive a low-energy effective field theory for a fully gapped, topologically ordered, fractionalized state with an eightfold ground-state degeneracy. This state is a fermionic symmetry-enriched topological phase with particle-number conservation and time-reversal symmetry. The topological terms in the effective field theory describe a quantized magnetoelectric response and nontrivial mutual braiding statistics of dynamical extended vortex loops with emergent fermions in the bulk. We explicitly compute the expected mutual statistics in a specific model on the pyrochlore lattice within a slave-particle mean-field theory. We argue that our model also provides a possible condensed-matter realization of oblique confinement.

Figure 1

(a) Pyrochlore lattice of corner-sharing tetrahedra. (b) Electronic band structure near the multicritical point $(t_{\sigma c},t_{\pi c})\approx (4.62,2.43)$ at $U=0$. The mean-field slave-fermion band structure is related by a uniform rescaling. The Fermi energy lies between the dispersive and flat bands. (c) $⟨{\tau }^x⟩$ as a function of $U$ at $(t_{\sigma },t_{\pi })=(0,0)$. The TI-TI* transition occurs at $U\approx 17$. (d) Slave-particle mean-field phase diagram in the $(t_{\sigma },t_{\pi })$ plane at $U=5.4$. The semimetallic phases ${\mathrm{SM}}_{\Gamma }$ and ${\mathrm{SM}}_L$ are distinguished by their Fermi points $\Gamma$ and $L$. The oval red line encloses the ${\mathbb{Z}}_2$ fractionalized phases (darker shade). Increasing $U$ expands these phases which emanate from the multicritical point.

Figure 2

(a) ${\mathbb{Z}}_2$ vortex loop (purple jagged line) on the pyrochlore lattice, threading hexagonal (above and in the kagome plane, shaded), and triangular plaquettes (below the kagome plane, shaded). Only a single kagome plane is shown. The (red) spheres and their relative size denote the probability density of a midgap state in the TI* phase that is localized on the loop. The axes employ units in which nearest-neighbor displacements are $2\sqrt{2/3}$ in length. (b) Accumulated Berry phase [54] for adiabatically braiding a slave fermion around a kagome plane hexagon pierced by the vortex loop (red dashed) and without the vortex loop (light blue). The difference of the two (solid black) yields the predicted statistical phase of $\pi$.