Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice from $t_{2g}$ Orbitals

The two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model on a honeycomb lattice has been shown to be a universal resource for quantum computation. In this valence bond solid, however, the spin interactions involve higher powers of the Heisenberg coupling $({\overrightarrow{S}}_i·{\overrightarrow{S}}_j{)}^n$, making these states seemingly unrealistic on bipartite lattices, where one expects a simple antiferromagnetic order. We show that those interactions can be generated by orbital physics in multiorbital Mott insulators. We focus on $t_{2g}$ electrons on the honeycomb lattice and propose a physical realization of the spin-$3/2$ AKLT state. We find a phase transition from the AKLT to the Néel state on increasing Hund’s rule coupling, which is confirmed by density matrix renormalization group simulations. An experimental signature of the AKLT state consists of protected, free $S=1/2$ spins on lattice vacancies, which may be detected in the spin susceptibility.

Figure 1

Nearest-neighbor $t_{2g}$ hoppings, corresponding to (a) $xy$, (b) $yz$, and (c) $zx$ orbitals, and (d) the resulting directional-hopping model, (e) free spins (denoted by stars) around lattice vacancy in the AKLT phase.

Figure 2

Phase diagram of the system Eq. (2) for $U^{\prime }<U$.

Figure 3

(a) Energy gap of model Eq. (3) in units of $J$ with and without an impurity. Inset: Finite size scaling of $E_{\mathrm{gap}}$ for $J_H/J=10$, (b) inverse correlation length ${\xi }^{-1}$ extracted from the exponential decay of $⟨s^z⟩$ [see Fig. 4].

Figure 4

(a) Expectation value $\sum _{\gamma }⟨s_{j,\gamma }^z⟩$ on one of the sublattices, along the cylinder, 8 hexagons in length. (b) Entanglement entropy $S_e$ of the ground state of model Eq. (3) normalized by the number of cut bonds.

Figure 5

Eigenvalues of the two-site Hamiltonian as a function of $J/J_H$. Inset above: two-site toy model.