Antiferromagnetic ordering of energy levels for a spin ladder with four-spin cyclic exchange: Generalization of the Lieb-Mattis theorem

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The Lieb-Mattis theorem is generalized to an antiferromagnetic spin-ladder model with four-spin cyclic exchange interaction. We prove that for $J>2\phantom{\rule{0.3em}{0ex}}\text{K}$, the antiferromagnetic ordering of energy levels takes place separately in two sectors, which remain symmetric and antisymmetric under the reflection with respect to the longitudinal axis of the ladder. We prove also that at the self-dual point $J=2\phantom{\rule{0.3em}{0ex}}\text{K}$, the Lieb-Mattis rule holds in the sectors with fixed number of rung singlets. In both cases, it agrees with the similar rule for Haldane chain with appropriate spin number.