Quantum loop states in spin-orbital models on the honeycomb lattice

The search for truly quantum phases of matter is a center piece of modern research in condensed matter physics. Quantum spin liquids, which host large amounts of entanglement—an entirely quantum feature where one part of a system cannot be measured without modifying the rest—are exemplars of such phases. Here, we devise a realistic model which relies upon the well-known Haldane chain phase, i.e. the phase of spin-1 chains which host fractional excitations at their ends, akin to the hallmark excitations of quantum spin liquids. We tune our model to exactly soluble points, and find that the ground state realizes Haldane chains whose physical supports fluctuate, realizing both quantum spin liquid like and symmetry-protected topological phases. Crucially, this model is expected to describe actual materials, and we provide a detailed set of material-specific constraints which may be readily used for an experimental realization.

In the orbital sector, the coupling Hamiltonian between two sites 1 and 2 connected by an x-type bond, as defined in the Methods, takes the form: In Eq. (1) of the main text, if J = 0, J 1 = −ζ, J 7 = −υ 1 , J 11 = −υ 2 , and all others zero. While the number of parameters is large (12), many of them are expected to be zero, physically. For example, it is unclear whether it is possible to obtain terms which involve a single power of angular momentum at each site from standard superexchange calculations [1,2].
Upon introducing the spin degrees of freedom, in principle, each independent coefficient may be a spin Hamiltonian of the form (for no spin-orbit coupling): where p = 1, .., 12 labels the independent terms. In Eq. (1) of the main text, we took J 1 = −ζ + J(S 1 · S 2 + β(S 1 · S 2 ) 2 ), i.e. A 1 = −ζ, B 1 = J, C 1 = Jβ, and J 7 = −υ 1 , i.e. A 7 = −υ 1 and B 7 = C 7 = 0, and J 11 = −υ 2 , i.e. A 11 = −υ 2 and B 11 = C 11 = 0 and all other terms zero. * lucile.savary@ens-lyon.fr Supplementary Fig. 1. Flippable plaquette adventure through third-order perturbation theory. The small hexagons next to the arrows show the bond on which H kin is applied at that order. The orbitals highlighted in purple do not have a matching orbital on the bond on which they lie.

Supplementary Note 2. Details of perturbation theory
Here we focus on orbital space, i.e. set J = 0, and give a few details for the degenerate perturbation theory in onto the manifold of loop coverings of the lattice, valid when ζ > 0 and ζ υ. The effective Hamiltonian is where H kin appears as many times as the order in perturbation theory.
Consider a "flippable" plaquette. Acting once with H kin on any bond which belongs to the plaquette creates two "defect" bonds (this configuration does not belong to the loop covering manifold), with the new plaquette state looking like on Supplementary Fig. 1. The energy of this configuration is that of a loop cut, i.e. ζ. Acting a second time with H kin , with the "active" bond operator one bond away from the first active bond creates another configuration of energy ζ. Only at third order is the system brought back to the loop manifold. There are twelve (= 6 × 2) ways to achieve this. It is noteworthy that including many other terms from Supplementary Eq. (1) will not produce a lower-order contribution.

Supplementary Note 3. Haldane chain energy
In this appendix we investigate the energy density (energy divided by the number of sites) of S = 1 loops in the Haldane phase as a function of their length. AKLT chains.-At the AKLT point, the energy density is independent of the loop length. Indeed the AKLT Hamiltonian may be rewritten as (5) i.e. as the sum of the projectors (with equal positive coefficient) onto the S tot = 2 sector (i.e. (S tot ) 2 = 2(2 + 1) = 6) of the S tot i = S i + S i+1 operator. This means that the ground state will have zero components in the S = 2 sector. Then, the energy is independent of chain length. Hence, at first order in perturbation theory, the spins do not lift the degeneracy of the loop coverings at the spin AKLT point. Numerical results away from the AKLT point.