Exact Results for a Spin-${\bf 1}$ lattice

Abstract

We consider a lattice of spin-1 particles with a general pairwise interaction $ [ {\rm cos} γ({\bf S}_{l} \cdot {\bf S}_{l+1}) \ + {\rm sin} γ({\bf S}_{l} \cdot {\bf S}_{l+1})^2 \ ]$. We show that, for a large class of lattices with even number of sites, the ground state for the region $ - {3 π\over 4} < γ< - {π\over 2}$ belongs to total spin $S_{\rm tot} = 0$, whereas the state of minimum excited energy but with finite $S_{\rm tot}$ belongs to $S_{\rm tot} = 2$. These results are constrasted with the generalized Marshall theorems, applicable to a bipartite lattice and $ - {π\over 2} < γ\le 0$.