Spin-Orbital Exchange of Strongly Interacting Fermions in the $p$ Band of a Two-Dimensional Optical Lattice

Mott insulators with both spin and orbital degeneracy are pertinent to a large number of transition metal oxides. The intertwined spin and orbital fluctuations can lead to rather exotic phases such as quantum spin-orbital liquids. Here, we consider two-component (spin $1/2$) fermionic atoms with strong repulsive interactions on the $p$ band of the optical square lattice. We derive the spin-orbital exchange for quarter filling of the $p$ band when the density fluctuations are suppressed, and show that it frustrates the development of long-range spin order. Exact diagonalization indicates a spin-disordered ground state with ferro-orbital order. The system dynamically decouples into individual Heisenberg spin chains, each realizing a Luttinger liquid accessible at higher temperatures compared to atoms confined to the $s$ band.

Figure 1

Virtual hopping processes giving rise to the spin-orbital exchange. $i$ and $j$ label two neighboring sites. An arrow in the upper (lower) semicircle means the $p_x$ ($p_y$) orbital is occupied by atoms of given spin. $a,b,c$ are intermediate states for two atoms on the same site $j$.

Figure 2

(a) The eigenstates of $H_x^i$ for a single bond. (b) One of the degenerate ground state of a two-leg ladder. The value of the nearest neighbor spin correlation is shown graphically.

Figure 3

(a) The ground state of $H_{\text{so}}$ for a $3\times 4$ cluster with periodic boundary conditions. (b) The ground state of a single chain ($1\times 4$) with periodic boundary condition in the $y$ direction. The ground state energy $E_g$ is measured in $J=t^2/U$.

Figure 4

The ground state energy per site $E_g/N$ in units of $J$ obtained by exact diagonalization of $H_{\text{so}}$ for different clusters. Finite size scaling yields $E_g/N=-4.159J$ (filled star) in the thermodynamic limit $N\to \infty$.