  We have developed a path integral Monte Carlo method for simulating helium
films and apply it to the second layer of helium adsorbed on graphite. We use
helium-helium and helium-graphite interactions that are found from potentials
which realistically describe the interatomic interactions. The Monte Carlo
sampling is over both particle positions and permutations of particle labels.
  From the particle configurations and static structure factor calculations, we
find that this layer possesses, in order of increasing density, a superfluid
liquid phase, a sqrt(7) x sqrt(7) commensurate solid phase that is registered
with respect to the first layer, and an incommensurate solid phases. By
applying the Maxwell construction to the dependence of the low-temperature
total energy on the coverage, we are able to identify coexistence regions
between the phases. From these, we deduce an effectively zero-temperature phase
diagram. Our phase boundaries are in agreement with heat capacity and torsional
oscillator measurements, and demonstrate that the experimentally observed
disruption of the superfluid phase is caused by the growth of the commensurate
phase. We further observe that the superfluid phase has a transition
temperature consistent with the two-dimensional value. Promotion to the third
layer occurs for densities above 0.212 atom/A^2, in good agreement with
experiment. Finally, we calculate the specific heat for each phase and obtain
peaks at temperatures in general agreement with experiment.
