Effective Hamiltonian and low-lying eigenenergy clustering patterns of four-sublattice antiferromagnets

We study the low-lying eigenenergy clustering patterns of quantum antiferromagnets with p sublattices ~in particular p54). We treat each sublattice as a large spin, and using second-order degenerate perturbation theory, we derive the effective ~biquadratic! Hamiltonian coupling the p large spins. In order to compare with exact diagonalizations, the Hamiltonian is explicitly written for a finite-size lattice, and it contains information on energies of excited states as well as the ground state. The result is applied to the face-centered-cubic Type-I antiferromagnet of spin 1/2, including second-neighbor interactions. A 32-site system is exactly diagonalized, and the energy spectrum of the low-lying singlets follows the analytically predicted clustering pattern.