Electronic and magnetic properties of the ionic Hubbard model on the striped triangular lattice at $\frac{3}{4}$ filling

We report a detailed study of a model Hamiltonian which exhibits a rich interplay of geometrical spin frustration, strong electronic correlations, and charge ordering. The character of the insulating phase depends on the magnitude of the onsite energy $\Delta /\left|t\right|$ and on the sign of the hopping amplitude $t$. We find a Mott insulator for $\Delta ⪢U⪢\left|t\right|$; a charge-transfer insulator for $U⪢\Delta ⪢\left|t\right|$; and a correlated covalent insulator for $U⪢\Delta \sim \left|t\right|$, with $U$ the onsite Coulomb repulsion energy. The charge-transfer insulating state is investigated using a strong-coupling expansion. The frustration of the triangular lattice can lead to antiferromagnetism or ferromagnetism depending on the sign of $t$. We identify the “ring” exchange process around a triangular plaquette which determines the sign of the magnetic interactions. Exact diagonalization calculations are performed on the model for a wide range of parameters and compared to the strong-coupling expansion. The regime $U⪢\Delta \sim \left|t\right|$ and $t<0$ is relevant to ${\text{Na}}_{0.5}{\text{CoO}}_2$. The calculated optical conductivity and the spectral density are discussed in the light of recent experiments on ${\text{Na}}_{0.5}{\text{CoO}}_2$.

Figure 1

(Color online) Spin and charge order in the 3/4-filled Ionic Hubbard model (1) on a triangular lattice in the limit $U⪢\Delta ⪢\left|t\right|$. $A$ and $B$ denote the inequivalent sites of the lattice. $C$-type antiferromagnetism (left) is found for $t>0$ in contrast to $G$-type antiferromagnetism (right) for $t<0$. The exchange couplings $J$ and $J_{\perp }$ are defined in Eq. (7) for the appropriate $t-J$ model in Eq. (6). A ferromagnetic exchange coupling, $J$, between neighboring $A$ sites occurs for the parameter range: $0<5t<\Delta <\sqrt{2U}$.

Figure 2

(Color online) Schematic phase diagram of the Ionic Hubbard model 1 on a striped triangular lattice at 3/4-filling. The transition lines are based on the lowest-order corrections in a strong-coupling analysis and on Lanczos diagonalization calculations. The $t<0$ case is relevant to the ${\text{Na}}_{0.5}{\text{CoO}}_2$ insulator. Insulating phases at strong-coupling, $U>W$, of different types are found ranging from a charge-transfer insulator (CTI), a Mott Insulator (MI), and a covalent insulator (CI). The bandwidth of the model for $\Delta =0$ (the isotropic triangular lattice), $W=9\left|t\right|$, is effectively reduced to $4\left|t\right|$ corresponding to one-dimensional (1D) chains as $\Delta$ increases. At exactly $\Delta =\infty$, the system is insulating for any nonzero $U$ as expected for a half-filled Hubbard chain due to Umklapp processes. The blue line is an estimate of the critical $U$ for the metal-to-insulator transition which follows the effective bandwidth dependence with $\Delta$. Insulating phases for $t<0$ display $G$-type antiferromagnetic (AFM) (see Fig. 1) correlations whereas a $C$-AFM region (see Fig. 1) for $t>0$ is obtained from the condition, $J<0$, to Eq. (7) which we assume valid for $U\gtrsim 9\left|t\right|$. The marked $\Delta =0$ axis for $t>0$ and above $U\approx 5\left|t\right|$ indicates the occurrence of ferromagnetism as predicted by DMFT of the Hubbard model on an isotropic triangular lattice. (Ref. 6)

Figure 3

Ionic Hubbard model on two- and four-site clusters. The energy-level diagram for three electrons on two sites (left) and the four-site cluster with six electrons (right).

Figure 4

(Color online) Dependence of ground-state energies with $\Delta$ for the four-site cluster with $t^{\prime }=t$ and $U=100\left|t\right|$. The Coulomb interaction $U$, $2U$, and $3U$ have been subtracted from the total energies $E_0\left(5\right)$, $E_0\left(6\right)$, and $E_0\left(7\right)$, respectively, for convenience.

Figure 5

(Color online) Dependence of the charge gap on $\Delta$ for the four-site cluster. Note the increase of ${\Delta }_c$ for any $\Delta$ for $U⪢\left|t\right|$ of Fig. 4 and also how the dependence for $\Delta >\left|t\right|$ is quite different with $\Delta <\left|t\right|$.

Figure 6

(Color online) Cluster shapes of different sizes used in exact diagonalization calculations.

Figure 7

(Color online) Charge disproportionation, $n_B-n_A$, between inequivalent rows in the ionic Hubbard model 1. In (a) tight-binding (dash-dotted lines) exact results for $U=0$ are compared with Lanczos diagonalization (closed symbols) for the $N_s=18$ tilted cluster of Fig. 6 showing good agreement. In (b) the dependence of $n_B-n_A$ with $U$ is shown from Lanczos diagonalization for $t<0$ on the same cluster.

Figure 8

(Color online) Difference in filling of the hybrid $\pm$ bands as a function of $\Delta /\left|t\right|$ for $t<0$ and several values of $U/\left|t\right|$. Results are from Lanczos calculations on 18-site clusters.

Figure 9

(Color online) Dependence of the charge gap on the on-site potential $\Delta$. The charge gap for $U=100\left|t\right|$ on a $N_s=18$ tilted cluster (a) and a $N_s=16$ ladder-type cluster (b) (see Fig. 6) is shown. Dashed lines denote the results of the strong-coupling expansion (19) for comparison.

Figure 10

(Color online) Development of magnetic order. The static spin structure factor $S\left(\vec{q}\right)$ at the wavevectors (a) ${\mathbf{Q}}_1=\left(0,\pi /\sqrt{3}\right)$, associated with $C$-type antiferromagnetism (Ref. 36) (see Fig. 1, left panel), and (b) ${\mathbf{Q}}_2=\left(\pi ,\pi /\sqrt{3}\right)$, associated with $G$-type antiferromagnetism (Ref. 36) (see Fig. 1, right panel), are shown for $U=100\left|t\right|$ as a function of $\Delta$, the difference between the local energies on the $A$ and $B$ sublattices. Calculations are performed on the $N_s=18$ cluster.

Figure 11

(Color online) Density of states of the charge-transfer insulator. We take $U=100\left|t\right|$, $\Delta =10\left|t\right|$, and $t<0$. The inset shows the low-energy part where the charge gap agrees with the strong-coupling expression Eq. (19). The excitation energies in the atomic limit $\left(t=0\right)$ are shown by the arrows below the abscissa. The chemical-potential $\left(\omega =\mu \right)$ is shown by the vertical dashed line.

Figure 12

(Color online) Evolution of the energy dependence of the density of states with decreasing charge transfer. We take $U=15\left|t\right|$ and vary $\Delta$ from $10\left|t\right|$ (top-left, charge-transfer insulator) to $\Delta =\left|t\right|$ (bottom, covalent insulator), all with $t<0$. The chemical-potential $\left(\omega =\mu \right)$ is shown by the vertical dashed lines.

Figure 13

(Color online) Frequency dependence of the optical conductivity for several values of $\Delta /t$ and fixed $U=15\left|t\right|$ with $t<0$.

Figure 14

Three-site “ring” exchange processes contributing to the exchange interaction, $J$, between neighbor spins in an $A$ chain of the $t-J-J_{\perp }$ model [compare Eq. (6)].

Figure 15

Four-site “ring” exchange processes of $\mathcal{O}({\left(t/\Delta \right)}^4\phantom{)}$ contributing to the exchange interaction between two neighboring sites in the $A$ chains in model (6).