Metal-Insulator Transition of Fermions on a Kagome Lattice at 1/3 Filling

We discuss the metal-insulator transition of the spinless fermion model on a kagome lattice at $\frac{1}{3}$ filling. The system is analyzed by using exact diagonalization, density-matrix renormalization group methods, and random-phase approximation. In the strong-coupling region, the charge-ordered ground state is consistent with the predictions of an effective model, i.e., plaquette order. We find that the qualitative properties of the metal-insulator transition are totally different depending on the sign of the hopping matrix elements, reflecting the difference in the band structure near the Fermi level.

Figure 1

Lattice structure of hydrogen-bonded crystals $M_3\mathrm{H}({\mathrm{XO}}_4{)}_2$ where the proton sites form the kagome lattice (right figure). The left figure gives an example of proton ordering below the critical temperature $T_c$. The solid (red) lines denote hydrogen bonds.

Figure 2

Band structure of kagome lattice fermions for positive (left) and negative (right) hopping energies.

Figure 3

Charge-ordering patterns in the $\frac{1}{3}$-filled kagome lattice characterized by wave vectors I $(q_a,q_b)=(0,0)$, II (0, $\pi$), and III ($2\pi /3$, $2\pi /3$). Circles on pattern III correspond to the cyclic hopping process, e.g., see Eq. (3).

Figure 4

(a) Finite clusters of the kagome lattice treated by the ED method. Broken lines indicate the geometry of the configurations of 12- and 18-site clusters used in the ED. From the 12- and 18-site clusters we obtain $E_{\mathrm{I}}-E_{\mathrm{II}}$ and $E_{\mathrm{II}}-E_{\mathrm{III}}$, respectively. (b) $L_x\times 3$ cluster (shaded regime) used for the DMRG method. Open and periodic boundary conditions are taken for the $x$ and $y$ directions, respectively.

Figure 5

(a) Energy differences $E_{\mathrm{II}}-E_{\mathrm{III}}$ for an 18 site cluster as a function of $1/V^2$ in units of $t$, and (c) $E_{\mathrm{I}}-E_{\mathrm{II}}$ for an 12 site cluster as a function of $1/V^3$. (c) Relationship of the energies $E_{\mathrm{I}}$, $E_{\mathrm{II}}$, and $E_{\mathrm{III}}$ corresponding to the CO patterns I–III in Fig. 3.

Figure 6

Single-particle gap $\Delta$ and order parameter $\eta$ as a function of nearest-neighbor repulsion $V$ for (a) positive and (b) negative $t$ values. Insets: finite-size scaling of each quantity for (a) $V=2$, 3, 4, and 5; and (b) $V=2$, 3, 4, 5, and 6, from bottom to top.

Figure 7

Phase diagram of the $\frac{1}{3}$-filled $t\mathrm{\text{−}}V$ model on the kagome lattice obtained by the random-phase approximation.