A Spin-1 Kagom\'{e} Heisenberg Antiferromagnet

We study a spin-1 Heisenberg antiferromagnet on a $2D$ kagom\'{e} lattice by projecting the Heisenberg Hamiltonian onto a restricted subspace of the full Hilbert space. This subspace consist of AKLT-like valence bound states described by closed loops. While not orthogonal, these singlet states are linearly independent; we derive the overlap between them and show that it is non-local and depends on the topology of nested loops. All of these states are characterized by the exponential decay of spin-spin correlations. Within this subspace, we identify lowest energy states which can be thought of as variational candidates for the ground states of the spin-1 kagom\'{e} Heisenberg and compare them with previous numerical studies.