  Quantum Monte Carlo and semiclassical methods are used to solve two and four
site cluster dynamical mean field approximations to the square lattice Hubbard
model at half filling and strong coupling. The energy, spin correlation
function, phase boundary and electron spectral function are computed and
compared to available exact results. The comparision permits a quantitative
assessment of the ability of the different methods to capture the effects of
intersite spin correlations. Two real space methods and one momentum space
representation are investigated. One of the two real space methods is found to
be significantly worse: in it, convergence to the correct results is found to
be slow and, for the spectral function, nonuniform in frequency, with
unphysical midgap states appearing. Analytical arguments are presented showing
that the discrepancy arises because the method does not respect the pole
structure of the self energy of the insulator. Of the other two methods, the
momentum space representation is found to provide the better approximation to
the intersite terms in the energy but neither approximation is particularly
acccurate and the convergence of the momentum space method is not uniform. A
few remarks on numerical methods are made.
