Ionic Hubbard model on a triangular lattice for ${\text{Na}}_{0.5}{\text{CoO}}_2$, ${\text{Rb}}_{0.5}{\text{CoO}}_2$, and ${\text{K}}_{0.5}{\text{CoO}}_2$: Mean-field slave boson theory

We present a strongly correlated mean-field theory of the ionic Hubbard model on the triangular lattice with alternating stripes of site energy using Barnes-Coleman slave bosons. We study the paramagnetic phases of this theory at three quarters filling, where it is a model of ${\text{Na}}_{0.5}{\text{CoO}}_2$, ${\text{Rb}}_{0.5}{\text{CoO}}_2$, and ${\text{K}}_{0.5}{\text{CoO}}_2$. This theory has two bands of fermionic quasiparticles: one of which is filled or nearly filled and hence weakly correlated; the other is half-filled or nearly half-filled and hence strongly correlated. Further results depend strongly on the sign of the hopping integral $t$. The light band is always filled for $t>0$, but only becomes filled for $\left|\Delta /t\right|\ge 1.5$ for $t<0$, where $\Delta$ is the difference in the site energies of the two sublattices. A metal-charge transfer insulator transition occurs at $\left|\Delta /t\right|=5.0$ for $t>0$ and $\left|\Delta /t\right|=8.0$ for $t<0$. In the charge transfer insulator complete charge disproportionation occurs: one sublattice is filled and the other is half-filled. We compare our results with exact diagonalization calculations and experiments on ${\text{Na}}_{0.5}{\text{CoO}}_2$ and discuss the relevance of our results to ${\text{Rb}}_{0.5}{\text{CoO}}_2$ and ${\text{K}}_{0.5}{\text{CoO}}_2$. We propose a resolution of seemingly contradictory experimental results on ${\text{Na}}_{0.5}{\text{CoO}}_2$. Many experiments suggest that there is a charge gap, yet quantum oscillations are observed suggesting the existence of quasiparticle states at arbitrarily low excitation energies. We argue that the heavy band is gapped while the light band, which contains less than one charge carrier per 100 unit cells, remains ungapped.

Figure 1

(Color online) The various unit cells and Brillouin zones discussed in this paper: (a) the unit cell of the isotropic triangular lattice; (b) the unit cell of the ionic Hubbard model [Eq. (2)]; and (c) the basal plane of the unit cell of $A_{0.5}{\text{CoO}}_2$; panels (d), (e), and (f) show the Brillouin zones corresponding to the unit cells in panels (a), (b), and (c), respectively. In (b) the different size circles distinguish the $A$ and $B$ sublattices. In (c) the large red (small blue) circles indicate the positions of Na ions above (below) the ${\text{CoO}}_2$ plane. The unit cell of the ionic Hubbard model (b) is twice as large as that of the Hubbard model of the triangular lattice (a). Further note that the basal plane of the unit cell of $A_{0.5}{\text{CoO}}_2$ is twice as large as the unit cell of the ionic Hubbard because of the periodicity of the (out of plane) $A$ ions. Selected high symmetry points are marked in the Brillouin zones. Note that the Brillouin zones are drawn to scale and that the $\Gamma$ point $\left[\mathbf{k}=\left(0,0\right)\right]$ is equivalent in each Brillouin zone.

Figure 2

(Color online) Charge disproportionation as a function of $\left|\Delta /t\right|$ for both signs of $t$. Here we compare the real-space charge disproportionation, $n_A-n_B$, calculated by exact diagonalization (Ref. 23) with the results from the slave boson calculations where $q_{-}=\left(n_A-n_B\right)/2$. The two methods are in excellent quantitative agreement for small $\left|\Delta /t\right|$, but at large $\left|\Delta /t\right|$ the slave boson theory predicts complete charge disproportionation; i.e., one sublattice becomes completely filled. In the exact diagonalization complete charge disproportionation is only approached asymptotically as $\left|\Delta /t\right|\to \infty$. In the slave boson theory the system is metallic except when there is complete charge disproportionation. In the exact diagonalization calculations a gap opens when there is only a small charge transfer between the two sublattices. We also plot the reciprocal space (band) charge disproportionation, $n_{-}-n_{+}$. We see that the MIT does not occur at the same $\left|\Delta /t\right|$ as the bonding band becomes filled, in spite of this implying that the antibonding band is half-filled and the fact that $U=\infty$. This emphasizes the fact that the half-filled band is not equivalent to a half-filled one-dimensional (1D) chain. We plot $q_{-}$ (similar results are found for other parameters) for two different sized $\left(L\times L\right)$ reciprocal space meshes, in the $L=1000$ case the convergence criteria is also tightened by a factor of 50, which demonstrates that the calculations are well converged.

Figure 3

(Color online) The ground-state energy, $F$, relative to that in the insulating state for the same value of $\Delta$ as a function of $q_{-}$ for various values of $\Delta$ and $t<0$. The solid black line indicates the free energy found at the self-consistent solution, with crosses marking the data for values of $\Delta$ at which we plot the full $F\left(q_{-}\right)$ curve. This plot indicates that the self-consistent solutions are indeed minima of $F$ with respect to the variation in $q_{-}$ [note that Eq. (16a) only requires that the self-consistent solution is a turning point in $F$ with respect to $q_{-}$, whereas, physically, we seek the minimum]. These results also strongly suggest that the metal-insulator transition in the slave boson theory is second order as no double-well structure is seen in $F\left(q_{-}\right)$.

Figure 4

(Color online) Same as Fig. 3 but for $t>0$. Again this shows that the self-consistent solutions are true minima and that the metal-charge ordered insulator transition is second order.

Figure 5

(Color online) Band structure of the fermions for $t<0$. The band structure is not unlike that of the noninteracting system (Fig. 14). However, for the fermions, increasing $\left|\Delta /t\right|$ has a much stronger effect than it does for the noninteracting electrons. In particular as $\left|\Delta /t\right|$ is increased the antibonding band (solid lines) is flattened and eventually becomes flat when the metal-insulator transition is reached at $\left|\Delta /t\right|=8.0$. The high symmetry points of the first Brillouin zone are shown in Fig. 1e. Also see the online movie (Ref. 56).

Figure 6

(Color online) Band structure of the fermions for $t>0$. There are significant differences from the band structure of $t<0$ (Fig. 5). In particular the metal-insulator transition occurs at only $\left|\Delta /t\right|=5.0$. Also see the online movie (Ref. 56).

Figure 7

(Color online) Density of states (in units where $\left|t\right|=a=1$) of the fermions for $t<0$. Dashed (blue and pink) lines indicate the contributions of the individual bands. The narrowing of the antibonding band, as $\left|\Delta /t\right|$ increases, is clearly visible from these plots, while it can be seen that the bonding band actually becomes a little wider as $\left|\Delta /t\right|$ increases. Clearly, the density of states (DOS) is very different from the noninteracting case, cf. Fig. 18, and these differences become greater as $\left|\Delta /t\right|$ increases. Also see the online movie (Ref. 56).

Figure 8

(Color online) Density of states (in units where $\left|t\right|=a=1$) of the fermions for $t>0$. Dashed (blue and pink) lines indicate the contributions of the individual bands. As in Fig. 7 we see that the antibonding band rapidly narrows as $\left|\Delta /t\right|$ increases. Furthermore, the width of the bonding band is almost independent of $\left|\Delta /t\right|$. Again, the DOS is very different from the noninteracting case, cf. Fig. 19, particularly for large $\left|\Delta /t\right|$. Also see the online movie (Ref. 56).

Figure 9

(Color online) Fermi surface of the fermions for $t<0$. For small $\left|\Delta /t\right|$ there are small Fermi pockets around the $Y$ points. Comparing the Fermi surface with the band structure of the fermions shows that these pockets arise from the bonding band. One, therefore, expects that these fermions are lighter than those in the quasi-one-dimensional sheets, which arise from the antibonding band (cf. Fig. 13). The Brillouin zone is that of the ionic Hubbard model [cf. Fig. 1e].

Figure 10

(Color online) Fermi surface of the fermions for $t>0$. For $t>0$ the bonding band is always completely filled. Therefore, this Fermi surface arises entirely from the antibonding band. As $\left|\Delta /t\right|$ is increased only rather subtle changes are observed in the Fermi surface, most notably, the nesting is slightly decreased. The Brillouin zone is that of the ionic Hubbard model [cf. Fig. 1e].

Figure 11

(Color online) Cut of the spectral density, $A\left(\mathbf{k},\omega \right)$, along the $\Gamma \text{−}M$ direction of the triangular lattice Brillouin zone [cf. Fig. 1d]. $A\left(\mathbf{k},\omega \right)$ is calculated from Eq. (18), broadened as described in the text below Eq. (23), and plotted for $k_{B}T=\left|t\right|/60\simeq 20\text{K}$. Green dotted lines are the slave boson dispersion; blue dashed lines are the bare dispersion. The abscissa is in ${\text{Å}}^{-1}$, with the lattice constant taken as $a=2.82\text{Å}$, as is appropriate for ${\text{Na}}_{0.5}{\text{CoO}}_2$ (Ref. 9). As $\left|\Delta /t\right|$ is increased the spectral weight is transferred from the antibonding band to the bonding band and the antibonding band narrows. Also see the online movie (Ref. 56).

Figure 12

(Color online) Cut of the spectral density, $A\left(\mathbf{k},\omega \right)$, along the $\Gamma \text{−}K$ direction of the triangular lattice Brillouin zone [cf. Fig. 1d]. $A\left(\mathbf{k},\omega \right)$ is calculated from Eq. (18), broadened as described in the text below Eq. (23), and plotted for $k_{B}T=\left|t\right|/60\simeq 20\text{K}$. Green dotted lines are the slave boson dispersion; blue dashed lines are the bare dispersion. The abscissa is in ${\text{Å}}^{-1}$, with the lattice constant taken as $a=2.82\text{Å}$, as is appropriate for ${\text{Na}}_{0.5}{\text{CoO}}_2$ (Ref. 9). At small $\left|\Delta /t\right|$ the bonding band crosses the Fermi energy in this direction (cf. Figs. 5, 9). As $\left|\Delta /t\right|$ is increased the bonding band is pushed below the Fermi energy and spectral weight is transferred from the antibonding band to the bonding band, which is not dispersive in this direction. Also see the online movie (Ref. 56).

Figure 13

(Color online) Variation in the renormalized cyclotron effective mass, $m_{\pm }$, in units of the free electron mass, $m$, with $\left|\Delta /t\right|$ for $t<0$. Note that in the main panel the ordinate is plotted on a logarithmic scale: the same data are plotted on a linear scale in the inset. $m_{-}/m\sim 1$ and $m_{+}/m\sim 4$ for small $\left|\Delta /t\right|$. The cyclotron effective mass of the antibonding band diverges as the metal-charge transfer insulator transition is approached consistent with our finding that the metal-charge ordered insulator transition is second order, cf. Fig. 3. The cyclotron effective masses are calculated from Eq. (24) with parameters appropriate for ${\text{Na}}_{0.5}{\text{CoO}}_2$: $t=-0.1\text{eV}$, an in-plane lattice constant, $a=2.82\text{Å}$, and a unit cell basal area of $\sqrt{3}a\times 2a$.

Figure 14

(Color online) Band structure of the ionic Hubbard model for $U=0$, at three-quarters filling, and $t<0$. Increasing $\left|\Delta /t\right|$ makes the system increasingly anisotropic, which serves to flatten the bands slightly in some directions. The constraint of three-quarters filling “pins” the antibonding band to the Fermi energy, but increasing $\left|\Delta /t\right|$ pushes the bonding band to lower energies.

Figure 15

(Color online) Band structure of the ionic Hubbard model for $U=0$, at three-quarters filling, and $t>0$. This band structure is markedly different from the $t<0$ case (Fig. 14). This is consistent with the general expectation that particle-hole symmetry is absent on frustrated lattices and hence that the sign of $t$ has strong effects on the physics.

Figure 16

(Color online) Fermi surface for noninteracting electrons for $t<0$. For small $\left|\Delta /t\right|$ there are small Fermi pockets around the $Y$ points. As $\left|\Delta /t\right|$ is increased the Fermi surface becomes increasingly quasi-one-dimensional.

Figure 17

(Color online) Fermi surface for noninteracting electrons for $t>0$. For $t>0$ the bonding band is always completely filled. Therefore, this Fermi surface arises entirely from the antibonding band. Note that the Fermi surface is very different from that for $t<0$ (Fig. 16).

Figure 18

(Color online) Density of states for $U=0$ and $t<0$. Dashed (blue and pink) lines indicate the contributions of the individual bands.

Figure 19

(Color online) Density of states for $U=0$ and $t>0$. Dashed (blue and pink) lines indicate the contributions of the individual bands.