Combined Topological and Landau Order from Strong Correlations in Chern Bands

We present a class of states with both topological and conventional Landau order that arise out of strongly interacting spinless fermions in fractionally filled and topologically nontrivial bands with Chern number $C=\pm 1$. These quantum states show the features of fractional Chern insulators, such as fractional Hall conductivity and interchange of ground-state levels upon insertion of a magnetic flux. In addition, they exhibit charge order and a related additional trivial ground-state degeneracy. Band mixing and geometric frustration of the charge pattern place these lattice states markedly beyond a single-band description.

Figure 1

CDW and FCI on the triangular lattice at $\nu =2/3$ corresponding to $\overline{n}=1/3$. (a) Phase diagram of model Eqs. (1a)–(1e) for a $3\times 3$ unit-cell ($6\times 3$ lattice-site) system with six fermions and $V_2=V_3=0$. (b) The $4\times 4-1=15$ unit-cell (30 lattice-site) cluster used in this Letter: the unit cell (black or red circles) and the charge-order pattern (filled circles) arising in the CDW and the hoppings. Solid, dashed, and arrowed black lines represent complex hoppings with phases of 0, $\pi$, and $\pm \pi /2$, respectively. Gray lines denote hoppings deactivated by the CDW. Third nearest-neighbor hoppings are not shown.

Figure 2

Spectral flow upon flux insertion for the model of Eqs. (1a) and (1e) for a 15 unit-cell system at (a) $\nu =12/15$ and (b) $\nu =13/15$. In panels (c)–(e) [(f)–(h)], the 15 low-energy states of (a)  [(b)] are divided into three groups of five; each group shows spectral flow consistent with a denominator-5 FCI. Fluxes are inserted as additional phases in the hoppings, totaling to $\phi$ for a loop around the cluster along direction ${\mathbf{a}}_3$. Parameters are $t^{\prime }/t=0.2$, $V_1/t=10$, and $V_2=V_3=2t$.

Figure 3

Charge-structure factor $N(\mathbf{k})$. (a) $\overline{n}=12/30$ and (b) $\overline{n}=13/30$, $V_2=V_3=2t$, and $V_1=8t$ ($+$), $V_1=10t$ ($\times$), and $V_1=12t$ (*). (c) Comparison of topologically trivial and nontrivial kinetic energy for $V_1/t=10$, $V_2=V_3=2t$. $t^{\prime }=0.2t$ and in the trivial case, $g_3=m=2t$ [47]. Of the 15 available momenta, numbers 5 and 10 correspond to the ordering momenta $\pm \mathbf{K}$ of the CDW.

Figure 4

Phase diagram for $\overline{n}=12/30$ depending on $V_1$ and $t^{\prime }$ for $V_2=V_3=2t$. Filled circles denote TPL states, $\times$ denotes states that do not show a gap between the lowest 15 states and the rest of the spectrum for all fluxes $\mathit{\phi }=({\phi }_{{\mathbf{a}}_2},{\phi }_{{\mathbf{a}}_3})$ [48].