  In prior work, the authors developed a method of degenerate perturbation
theory about the Ising limit to derive an effective Hamiltonian describing
quantum fluctuations in a half-polarized magnetization plateau on the
pyrochlore lattice. Here, we extend this formulation to an arbitrary lattice of
corner sharing simplexes of $q$ sites, at a fraction $(q-2k)/q$ of the
saturation magnetization, with $0<k<q$. We present explicit effective
Hamiltonians for the examples of the checkerboard, kagome, and pyrochlore
lattices. The consequent ground states in these cases for $k=1$ are also
discussed.
