Ordering Energy Levels of Interacting Spin Systems

Abstract

The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) ] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S ≥ 0. In ferrimagnetism, E(S + 1) > E(S) for $\mathrm{S\ge }\mathcal{S}$ , and in addition the ground state belongs to $\mathrm{S\le }\mathcal{S}$ . $\mathcal{S}$ is defined as follows: Let the maximum spin of the A sublattice be S_A and of the B sublattice S_B; then $\mathcal{S}{\mathrm{\equiv |S}}_A{\mathrm{-S}}_B|$ . Antiferromagnetism is treated as the special case of $\mathcal{S}\text{=0}$ . We also briefly discuss the structure of the lowest eigenfunctions in an external magnetic field.