Exact Results for a Spin-$1$ lattice

Abstract

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

to appear in J. Phys. Cond. Matt