The Bose--Hubbard Model and Superfluid Staircases in ⁴He Films

The recent experimental observation of step-like structure in the superfluid density of $^{4}$He films on graphite is interpreted in terms of passage close to Mott insulating lobes in the Bose--Hubbard model. Plateaus develop in the superfluid density near the completion of each layer because of the He-He repulsion. We modify the Bose--Hubbard Hamiltonian, using a density-dependent He-He interaction term to model the He-substrate potential. Quantum Monte Carlo simulations yield results which agree well with the experimental data.