Cooperative orbital ordering and Peierls instability in the checkerboard lattice with doubly degenerate orbitals

It has been suggested that the metal-insulator transitions in a number of spinel materials with partially filled $t_{2g}$ $d$ orbitals can be explained as orbitally driven Peierls instabilities. Motivated by these suggestions, we examine theoretically the possibility of formation of such orbitally driven states within a simplified theoretical model, a two-dimensional checkerboard lattice with two-directional metal orbitals per atomic site. We include orbital ordering and interatom electron-phonon interactions self-consistently within a semiclassical approximation, and onsite intraorbital and interorbital electron-electron interactions at the Hartree-Fock level. We find a stable, orbitally induced Peierls bond-dimerized state for carrier concentration of one electron per atom. The Peierls bond distortion pattern continues to be period two bond dimerization even when the charge density in the orbitals forming the one-dimensional band is significantly smaller than 1. In contrast, for carrier density of half an electron per atom the Peierls instability is absent within one-electron theory as well as mean-field theory of electron-electron interactions, even for nearly complete orbital ordering. We discuss the implications of our results in relation to complex charge, bond, and orbital ordering found in spinels.

Figure 1

(Color online) (a) 2D checkerboard lattice with orbital-ordering pattern given by the $H_{\text{OO}}$ term in Eq. (3). Filled dots denote metal atom positions, with two orbitals per atom, $xz$ and $yz$. Hopping terms are included along $x$, $y$, $x+y$, and $y-x$ directions as defined in Eqs. (5, 6, 7, 8). Hopping along the $t_x$ and $t_y$ directions, indicated by dashed lines, is modulated by the interatomic e-p coupling. (b) The $\left(dd\pi \right)$ overlap of $xz$, shown in green (light gray), and $yz$, shown in blue (dark gray), along the $x$ and $y$ directions. Note that along these directions, nonzero hopping matrix elements exist only along $x$ $\left(xz-xz\right)$ or $y$ $\left(yz-yz\right)$ direction. (c) The $\left(dd\pi \right)$ overlap of $xz$ and $yz$ orbitals along the $y-x$ direction. The overlap along $x+y$ is similar apart from a change of sign for the interorbital terms.

Figure 2

(Color online) Pattern of the orbital-ordering and bond-order modulation for one electron per atom. Squares (circles) represent atoms with predominant $xz$ $\left(yz\right)$ orbital occupation. Bonds alternate in strength along diagonal directions with solid (dashed) lines indicating strong (weak) bonds.

Figure 3

(Color online) Cooperative orbital-ordering and Peierls bond alternation for one electron per atom, with $U=U^{\prime }=0$. Calculations are for a $16\times 16$ periodic lattice. (a) Majority charge density (see text) in the orbitals forming the quasi-1D chains following orbital ordering. Here and in (b) circles, diamonds and squares correspond to $g=0.6$, 0.8, and 1.0, respectively. (b) $\Delta B$ along the diagonal chain directions (see text and Fig. 2) as a function of $\alpha$. (c) Bond alternation plotted as a function of charge density. Circles show the effect of increasing $g$ with constant $\alpha =1.3$. Diamonds show effect of increasing $\alpha$ with constant $g=0.8$. Lines are guides to the eye.

Figure 4

Majority charge density as a function of $U$ and $U^{\prime }$ for one electron per atom, normalized with respect to the same quantity for the uncorrelated system. Results shown are for $16\times 16$ periodic lattices with $g=0.8$, $\alpha =1.2$, and $K_{\text{OO}}=K_{\text{SSH}}=1$. Circles are for $U^{\prime }=0.3U$, diamonds are for $U^{\prime }=0.5U$, and triangles are for $U^{\prime }=0.7U$. Lines are guides to the eye.

Figure 5

(Color online) Cooperative orbital-ordering and Peierls bond alternation for 0.5 spinless fermions per atom. Calculations are for a $16\times 16$ periodic lattice. (a) Majority charge density in the orbitals forming the 1D chains. Here and in (b) circles, diamonds and squares correspond to $g=0.6$, 0.8, and 1.0, respectively. (b) $\Delta B$ along the diagonal chain directions (see Fig. 2) as a function of $\alpha$. (c) Bond alternation versus majority charge density. Circles show the effect of increasing $g$ with constant $\alpha =1.3$. Diamonds show effect of increasing $\alpha$ with constant $g=0.8$. Lines are guides to the eye.