Interaction-driven topological insulators on the kagome and the decorated honeycomb lattices

We study the spinless and spinful extended Hubbard models with repulsive interactions on the kagome and the decorated honeycomb (star) lattice. Using Hartree-Fock mean-field theory, we show that interaction-driven insulating phases with nontrivial topological invariants (Chern number or $Z_2$ invariant) exist for an experimentally reasonable range of parameters. These phases occur at filling fractions which involve either Dirac points or quadratic band crossing points in the noninteracting limit. We present comprehensive mean-field phase diagrams for these lattices and discuss the competition between topologically nontrivial phases and numerous other ordered states, including various charge, spin, and bond orderings. Our results suggest that $Z_2$ topological insulators should be found in a number of systems with either little or no intrinsic spin-orbit coupling.

Figure 1

(Color online) The spinless flux pattern developed by nearest- and second-nearest interactions that preserves lattice symmetry but spontaneously breaks time-reversal symmetry on the kagome lattice. Charges are uniform on all sites. The solid (blue) and the dashed (red) lines represent nearest-neighbor and second nearest-neighbor complex hopping, respectively. For the spinful case, two copies of the same (opposite) flux patterns for spin-up and spin-down fermions form the quantum anomalous Hall (topological insulator) state.

Figure 2

(Color online) Three different charge-density wave patterns on the kagome lattice studied in this paper. They are characterized by the wave vectors ${\mathit{q}}_{\text{I}}=\left(0,0\right)$, ${\mathit{q}}_{\text{II}}={\mathit{b}}_2/2$, and ${\mathit{q}}_{\text{III}}=\left({\mathit{b}}_1-{\mathit{b}}_2\right)/3$. Blue sites stand for the fermion-rich (poor) sites at 1/3 (2/3) filling fraction and white sites for the fermion-poor (rich) sites at 1/3 (2/3) filling fraction. The bond expectation values oscillate in the real space as well. We distinguish strong and weak bonds by thick and thin lines. For simplicity, we do not show the second-neighbor bonds.

Figure 3

The phase diagram of the extended Hubbard model for spinless fermions at 1/3 filling fraction on the kagome lattice. The third neighbor interaction is (a) $V_3=0$ and (b) $V_3=0.4t$. SM denotes the semimetallic phase with two Dirac points, QAH denotes a time-reversal symmetry broken quantum anomalous Hall phase and CDW I and III are charge density waves with patterns shown in Fig. 2. Solid lines denote first and dashed lines second-order transitions.

Figure 4

The phase diagram of the extended Hubbard model for spinless fermions at 2/3 filling on the kagome lattice. The dotted line indicates where the gap opens in the CDW I phase. For $V_2=0$, CDW I and II coexist in the gray region and CDW III in the black region. We set $V_3=0$.

Figure 5

(Color online) The fluxes ${\Phi }_{1,2,3}$ through elementary triangles in the QAH phase at filling fraction $f=2/3$. These elementary triangles form the unit cell, as shown in the inset. Because of the periodic boundary conditions on the unit cell, the net flux is zero and the individual fluxes satisfy $2{\Phi }_1+{\Phi }_2+3{\Phi }_3=0$. We have set $V_2=V_1/2$ and $V_3=0$.

Figure 6

(Color online) (a) the CDW order parameter $\rho$ (b) the bond order defined as the difference between two nearest strong and weak bonds (c) the gap at 2/3 filling fraction on the kagome lattice. We have set $V_2=t$. (d) the splitting of a QBCP (red circle) into two Dirac points (blue cross) for gapless CDW I.

Figure 7

(Color online) Schematic of four types of candidate phases on the kagome lattice for 1/3 filling fraction: (a) SCDW I (b) SCDW II (c) bond-order wave (BOW) and (d) ferromagnet (FM). Upward arrows and downward arrows denote the magnetization on each site. The same/different circles represent same/different numbers of fermions on corresponding sites. For simplicity we only show nearest bonds (the addition of two spin-resolved bonds) and do not show the second-nearest bonds. Stronger bonds are shown in bold.

Figure 8

The phase diagram of the spinful model at 1/3 filling on the kagome lattice. The SCDW I and II phase involve both a finite charge and spin-density-wave order parameter. Furthermore, when $U$ competes with $V_1$ a BOW is found. Solid lines indicate first order and dashed lines second-order transitions.

Figure 9

The $U\text{−}{V}_2$ phase diagram for $V_1=0$ and $V_3=0.4t$. An interaction-driven TI appears for finite $U$ and $V_2$. Solid lines indicate first-order and dashed lines second-order transitions

Figure 10

The $U\text{−}{V}_1$ phase diagram for $V_2=V_3=0$ at 2/3 filling fraction. Similar to the spinless case at 2/3 filling fraction, CDW I has nodes which separates itself from gapped phase by a dashed-dotted line. Similarly, SCDW I has nodes for small interaction strengths.

Figure 11

(Color online) (a) the flux pattern developed by interactions that preserves the lattice symmetry but spontaneously breaks time-reversal symmetry on decorated honeycomb (star) lattice. The blue solid line with an arrow (two arrows) represents a nearest-neighbor intratriangle(intertriangle) complex hopping while the red dashed line with an arrow represents the second-neighbor complex hopping. (b) the favorable CDW pattern from solutions of self-consistency equations. We have used different combinations of color and markers to show the mirror symmetry. Real first-neighbor bonds consistent with symmetry of the CDW pattern have been assumed (second-neighbor bonds not shown).

Figure 12

The phase diagram of the extended Hubbard model of spinless fermions at 1/2 filling on the decorated honeycomb lattice. Due to the mutual frustration of $V_1$ and $V_2$, the QAH phase occupies the middle part of phase diagram and the regions of CDW phase have been split into two parts. We have set $t^{\prime }=t$.