Quarter-filled Kitaev-Hubbard model: A quantum Hall state in an optical lattice

We analyze the physics of cold atoms in honeycomb optical lattices with on-site repulsion and spin-dependent hopping that breaks time-reversal symmetry. Such systems, at half filling and large on-site repulsion, have been proposed as a possible realization of the Kitaev model. The spin-dependent hopping breaks the spin degeneracy and, if strong enough, leads to four nonoverlapping bands in the noninteracting limit. These bands carry nonzero Chern number and therefore the noninteracting system has nonzero angular momentum and chiral edge states at 1/4 and 3/4 filling. We have investigated the effect of interactions on a quarter-filled system using the variational cluster perturbation theory and found that the critical spin-dependent hopping that separates the metal from the quantum Hall state is affected by interactions.

Figure 1

The honeycomb lattice, the NN vectors ${\mathbf{b}}_a$, and the lattice basis ${\mathbf{e}}_{1,2}$. The A and B sublattices are represented by green and blue dots, respectively.

Figure 2

The dispersion relation ${\epsilon }_{\pm }\left(\mathbf{k}\right)$ along a high-symmetry path in the Brillouin zone. The negative energy bands are symmetrically located underneath the horizontal axis and do not touch the upper bands. $K$ is the Dirac point, and $M=(K+K^{\prime })/2$ is the midpoint between the two Dirac points. Note the new Dirac point at $t^{\prime }=0.5$, about midway between $K$ and $M$. At $t^{\prime }=0$, ${\epsilon }_{+}$ and ${\epsilon }_{-}$ are degenerate.

Figure 3

The spectrum for an open tube of circumference $L=60$ with zig-zag edges, for $t^{\prime }=t$. Note the edge states that cross the gaps between the bands.

Figure 4

Orbital magnetization of the model as a function of the filling $n$ for $t^{\prime }=0.5$,1.0.

Figure 5

Clusters used in this work. (Left) Six-site cluster used for the two-dimensional model. (Right) Ten-site cluster used for one-dimensional zigzag ribbons. Intercluster links are represented by dashed lines and the repeated unit is shaded in gray.

Figure 6

The phase diagram of the model in U-$t^{\prime }$ plane. The four black squares correspond to the spectral plots of Fig. 7.

Figure 7

Spectral function $A(\mathbf{k},\omega )$, as a function of $\omega$, for spin-up particles and wave vectors along high-symmetry directions at $t^{\prime }=2/3$, quarter filling, and $U=0,0.5,1$, and 2. The energy unit is set by $t=1$.

Figure 8

(Top) Single-particle spectral function of zigzag ribbons of 10 sites at $t^{\prime }=t=1$, as a function of energy $\omega$. (Top) Up spins; bottom, down spins. The red arrows indicate that the chiral edge states intersect the Fermi level for both the noninteracting and interacting systems