Effective Hamiltonians for some highly frustrated magnets

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.