Narrowing of Topological Bands due to Electronic Orbital Degrees of Freedom

The fractional quantum Hall effect has been predicted to occur in the absence of magnetic fields and at high temperature in lattice systems that have flat bands with a nonzero Chern number. We demonstrate that orbital degrees of freedom in frustrated lattice systems lead to a narrowing of topologically nontrivial bands. This robust effect does not rely on fine-tuned long-range hopping parameters and is directly relevant to a wide class of transition-metal compounds.

Figure 1

(a) Chiral spin ordering on the triangular lattice. (b) Spins forming a regular tetrahedron, numbers refer to sites in (a). (c) Flux-phase state on the kagome lattice. The unit cell is indicated by the dashed hexagon and the gauge choice by arrows; a flux $\varphi$ threads each triangle. (d) Nearest-neighbor hopping geometry in lattices with triangular symmetry. Grey lines illustrate the oxygen octahedra, black front facets illustrate the triangular geometry. Thick dotted, dashed, and solid lines indicate the bonds corresponding to the hopping matrices ${\widehat{T}}_1$, ${\widehat{T}}_2$, and ${\widehat{T}}_3$. Two $d_{3z^2-r^2}$ (top) and $d_{x^2-y^2}$ (bottom) orbitals are also shown.

Figure 2

Top: kagome lattice, bottom: triangular lattice. Left: bands of a strip geometry, clearly showing the chiral edge states of the flattened bands. Right: off-diagonal Hall conductivity as function of chemical potential, providing the Chern numbers for the flat bands. For the Kagome lattice $t^{\prime }=-0.46$, $\Delta =2.75$, $\varphi =\pi /4$ and for the triangular case $t^{\prime }=-0.46$, $\Delta =2.5$.

Figure 3

(a) The smallest band gap over bandwidth ratios for $e_g$ electrons the triangular lattice, for the gaps $M_{+}$ and $M_{-}$ indicated in Fig. 2a. (b) The same for $t_{2g}$ electrons on a triangular lattice and (c) for $e_g$ on the kagome lattice. Orbital splittings are indicated by $\Delta$.

Figure 4

Lowest energies of the interacting $e_g$ system on the triangular-lattice, depending on flux inserted for (a) $U=2$, $V=1$, and (b) $U=4$, $V=2$. The ED results presented here were obtained for a $2\times 6$ system with $t^{\prime }/t=-0.46$ and $\Delta =1.5$; see 36.