Fractional Chern insulator on a triangular lattice of strongly correlated $t_{2g}$ electrons

We discuss the low-energy limit of three-orbital Kondo-lattice and Hubbard models describing $t_{2g}$ orbitals on a triangular lattice near half-filling. We analyze how very flat single-particle bands with nontrivial topological character, a Chern number $C=\pm 1$, arise both in the limit of infinite on-site interactions as well as in more realistic regimes. Exact diagonalization is then used to investigate an effective one-orbital spinless-fermion model at fractional fillings including nearest-neighbor interaction $V$; it reveals signatures of fractional Chern insulator (FCI) states for several filling fractions. In addition to indications based on energies, e.g., flux insertion and fractional statistics of quasiholes, Chern numbers are obtained. It is shown that FCI states are robust against disorder in the underlying magnetic texture that defines the topological character of the band. We also investigate competition between a FCI state and a charge density wave (CDW) and discuss the effects of particle-hole asymmetry and Fermi-surface nesting. FCI states turn out to be rather robust and do not require very flat bands, but can also arise when filling or an absence of Fermi-surface nesting disfavor the competing CDW. Nevertheless, very flat bands allow FCI states to be induced by weaker interactions than those needed for more dispersive bands.

Figure 1

(a) Illustration of a triangular-lattice plane built of edge-sharing oxygen octahedra. (b) The five $d$ orbitals of the transition-metal ion in the center are split into an $e_g$ doublet and a $t_{2g}$ triplet due to the local cubic symmetry; the latter is further split into one $a_{1g}$ state and an $e_g^{\prime }$ doublet. (The splitting between the latter is exaggerated here for visibility.)

Figure 2

Illustration of chiral spin pattern and effective model including the magnetic texture as phase factors in hoppings.

Figure 3

Flat lower chiral subband in the Kondo-lattice model with infinite Hund's rule coupling (double-exchange model). (a) Shows the one-particle energies of three $t_{2g}$ orbitals coupled to localized spins, where the latter form a spin-chiral phase on a triangular lattice,[14, 42, 43, 44] see Fig. 2. The system is a cylinder, i.e., periodic boundary conditions along $y$ direction and open boundaries along $x$. The horizontal axis is the momentum in the direction with periodic boundaries. The gaps ${\Delta }_{\mathrm{JT}}$ and ${\Delta }_c$ denote the gaps due to crystal-field splitting $E_{\mathrm{JT}}$ and to the chiral spin state. (b) shows the figure of merit $M$, see Eq. (10), for the lower $a_{1g}$ subband. The curves for crystal-field splittings $E_{\mathrm{JT}}=4,4.5$, and 5 were already given in Fig. 3b of Ref. 22 and are repeated here for convenience.

Figure 4

Figure of merit $M$, see Eq. (10), for finite Hund's-rule coupling $J_{\mathrm{Kondo}}/t_0$ and $E_{\mathrm{JT}}=6t_o$. The bands designated as “upper” and “lower” refer to the two subbands of the $a_{1g}$ states with spin parallel to the localized spin, which are separated by the gap opening in the spin-chiral phase, see Fig. 3a.

Figure 5

(a) Energy dispersion $e_{\mathbf{k}}$ for $t^{\prime }/t=0.2$. The inset shows the path taken through the first Brillouin zone. (b) Flatness ratio $M$, see Eq. (10), as a function of $t^{\prime }/t$. The flatness ratio has been calculated from the dispersion along the high-symmetry directions shown in the inset on the left.

Figure 6

Eigenvalue spectrum and evolution of selected levels under flux insertion for different values of $V/t$. In all panels, $t^{\prime }/t=0.2$.

Figure 7

Eigenvalue spectrum and evolution of selected levels under flux insertion for different values of $t^{\prime }/t$. In all panels, $V/t=1$.

Figure 8

Eigenenergy spectrum and spectral flow for FCI state with quasiholes introduced (a) by enlarging the system or (b) by removing particles. The numbers of levels below the legends are (a) 35 and (b) 40, respectively, in agreement with the counting rule of Ref. 31. In the $7\times 2$-unit cell system, the filling fraction is $\nu =2/7$. There are seven quasidegenerate ground states (marked in red), which are slightly separated from the rest of the states in the low-energy sector and exhibit spectral flow (shown in different colors). Their Chern number is $C=2/7$ within the numerical error margin.

Figure 9

(a)–(c) Berry curvatures ${\mathcal{B}}_{i_{\mathbf{k}}}$, (d)–(f) their standard deviations $\mathfrak{S}\left({\mathcal{B}}_{i_{\mathbf{k}}}\right)$ from the exact average and relative deviations of Hall conductivities $\delta {\sigma }_{i_{\mathbf{k}}}/\nu$ as a function of grid size for the state $i_{\mathbf{k}}=6$, when it is in the excited-state quasicontinuum (top) and in the ground-state manifold (middle and bottom). The integrations in Eq. (26) have been approximated by simple Riemann sums. $N_{\varphi }$ is the number of ${\varphi }_{x/y}$ points taken in the range $[0,6\pi )$ in each direction.

Figure 10

Hall conductivity, see Eq. (26), for each of the three quasidegenerate FCI states as a function of (a) the interaction strength $V/t$ and (b) the third-neighbor hopping $t^{\prime }/t$ (right). Note the different scales on the vertical axes.

Figure 11

(a) and (b) Hall conductivities of the three lowest-energy states and their average as a function of disorder phase $\phi$. (c) and (d) Standard deviation $\mathfrak{S}\left(\mathcal{B}\right)$ of total Berry curvature $\mathcal{B}$, defined by adding up the many-body Berry curvatures ${\mathcal{B}}_n$ of the three lowest-energy states for each value of ${\varphi }_{x/y}$, and relative deviation $\delta \sigma /\nu$ of Hall conductivity as a function of grid size. $N_{\varphi }$ is the number of ${\varphi }_{x/y}$ points taken in the range $[0,2\pi )$ in each direction. The results shown in the right panels are for $\phi \simeq 2\pi /3$, but are qualitatively the same for all other values of $\phi$.

Figure 12

Dispersion (gray-/ color scale) and Fermi surface (FS) for $\nu =2\overline{n}=2/3$ (thick solid line) for (a) $t^{\prime }/t=0.16$ and $M=11.16$, (b) $t^{\prime }/t=0.19245$ and $M=24$ (very close to maximal $M$) and (c) $t^{\prime }/t=0.25$ and $M=5.5$. Dashed lines indicate the first Brillouin zone. (d) illustrates the CDW possible for $\nu =2\overline{n}=2/3$. Black and red/gray circles indicate the two sublattices of the spin-chiral order and the effective model Eq. (23), filled circles the particles in the CDW state induced by large $V/t$.

Figure 13

Gap of threefold degenerate FCI ground state as a function of $M$ and $V/t$, for (a) $\nu =1/3$ and (b) $\nu =2/3$, shown in color code. The dashed lines indicate the phase boundary, defined by the condition that the gap remains open for all values of inserted magnetic flux. The flatness ratio $M$ of the flat band of $H_{\mathrm{kin}}$ (bottom scale) is adjusted by varying $t^{\prime }/t$ (top scale). The maximum value of the flatness ratio ($M\simeq 24$) is marked by the dotted lines.

Figure 14

Static structure factor for various values of $t^{\prime }/t$ and $V/t$ at filling fractions (a) and (b) $\nu =1/3$ and (c)–(f) $\nu =2/3$. The sharp peaks indicate the formation of a charge-ordered state. In the FCI regime, the static structure factor of all three quasidegenerate ground states is identical.