We study the low-lying eigenenergy clustering patterns of quantum antiferromagnets with p sublattices (in particular p = 4). 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.
Zhang, N. G., Henley, C. L., Rischel, C., & Lefmann, K. (2002). Effective Hamiltonian and low-lying eigenenergy clustering patterns of four-sublattice antiferromagnets. Physical Review B Condensed Matter, 65(6), 064427.1-064427.17. https://doi.org/10.1103/PhysRevB.65.064427