As I have posted previously finding a realistic Heisenberg spin model Hamiltonian which has a spin liquid ground state has proven to be difficult. The model on the Kagome lattice was thought to be a prime candidate for many years, partly because the classical model has an infinite number of degenerate ground states. However, a few years ago Rajiv Singh and David Huse performed a series expansion study which suggested that the ground state was actually a valence bond crystal with a unit cell of 36 spins. In the picture below the blue lines represent spin singlets, and H, P, and E, denote Hexagons, Pinwheels, and Empty triangles respectively. This result was confirmed by my UQ colleagues Glen Evenbly and Guifre Vidal using a completely different numerical method based on entanglement renormalisation.
However, there are new numerical results using DMRG which appeared on the arXiv this week, by Yan, Huse, and White. They find a spin liquid ground state, with a gap to both singlet and triplet excitations.
I thank Ashvin Vishvanath for bringing this preprint to my attention.