Figure 1

Triangular perovskite lattice and $t_{2g}$ orbitals. Oxygen octahedra are indicated by lines, with black lines illustrating the front facets. Thick dotted (dashed, solid) lines indicate nearest-neighbor bonds along lattice vector ${\mathbf{a}}_1$ (${\mathbf{a}}_2$, ${\mathbf{a}}_3$). (a) Shows two $d_{xy}$ orbitals (top) and one $d_{xz}$ and $d_{yz}$ orbital (bottom). In (b), the orbitals reflecting the threefold lattice symmetry are shown: The two $e_g^{\prime }$ orbitals (bottom), which differ by their complex phases, will turn out to be half filled, while the $a_{1g}$ orbital (pointing out of the plane, see top) forms nearly flat bands with nontrivial topological character that can support spontaneous FQH states.Reuse & Permissions

Figure 2

Spin-chiral magnetic phase with topologically nontrivial bands stabilized by onsite Coulomb interactions in $t_{2g}$ electrons on a triangular lattice. (a) Chiral magnetic order, the sites of the unit cell are labeled by 1 to 4. (b) The spins on the four sites can be seen as pointing to the corners of a tetrahedron; i.e., the pattern is noncoplanar. (c) One-particle energies on a cylinder (periodic boundary conditions along $x$) in the mean-field [35] ground state of the $t_{2g}$ multiorbital Hubbard model, which is given by the pattern shown in (a). States drawn in black (gray) have more (less) than 33% $a_{1g}$ character, dashed and dotted lines indicate edge states with more than 33% of their weight on the top (bottom) row of sites. The arrows $\uparrow$ ($\downarrow$) indicate states with electron spin mostly (anti-)parallel to the local quantization axis, which can be seen as the lower (upper) Hubbard band. The filling is 2.5 electrons per site, slightly less than half filling. Parameters used were $t=1$, $t_{dd}=0$, $U/t=12$, $J/t=3$, ${\Delta }_{\mathrm{JT}}/t=-6$. The figure of merit $M$, which is given by the ratio of the gap separating the two $a_{1g}$ subbands of the lower Hubbard band and the band width of the highest subband of the lower Hubbard band, is $M\approx 14$.Reuse & Permissions

Figure 3

Stability of the spin-chiral phase and flatness of the topological bands depending on parameters of the Hamiltonian. In (a), shaded areas in the $t_{dd}\mathrm{\text{−}}{\Delta }_{\mathrm{JT}}$ plane indicate a spin-chiral ground state Fig. 2a, 2b for $U/t=12$, white areas have a different ground state. Shading indicates the figure of merit $M$ for the flatness of the upper chiral subband, bright thick lines bound the region with $M\ge 10$. (b) shows $M$ depending on $t_{dd}$ for selected sets of $U$ and ${\Delta }_{\mathrm{JT}}$. Where the “Mott gap”, which separates the flat topologically nontrivial band from the upper Hubbard band, becomes very small, $M$ is determined by the minimal gap separating the band of interest from other bands. $J=U/4$ and $t=1$ were used in all cases.Reuse & Permissions

Figure 4

FQH state induced by NN Coulomb repulsion $V$ in the effective one-band model Eq. (1). (a) Energy depending on total momentum $\mathbf{k}$ for several values of $V/t$. (b) Energy for $V/t=0.2$ depending on a flux ${\varphi }_y$ added whenever an electron goes once around the whole lattice in $y$ direction. Each addition of $\varphi =2\pi$ leads to an equivalent state, $6\pi$ to the same state. The Chern numbers associated with the three low-energy states are almost exactly $2/3$ for $V/t=0.2$. Lattice size is $4\times 6$ sites (12 two-site unit cells), parameters in Eq. (1) are $t_1=0.27t$ and $t_3=-0.06t$, giving bands with $M\approx 13$ and a gap of $0.89t$. The filling of the flat band is $2/3$.Reuse & Permissions