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

TL;DR

This paper derives exact ground state and excitation properties for a spin-1 lattice with specific interactions, revealing spin configurations and energy levels for certain interaction parameters, and contrasting with known theorems.

Contribution

It provides exact results for the ground state and excited states of a spin-1 lattice with a general pairwise interaction, extending understanding beyond bipartite lattices.

Findings

Ground state has total spin S_{tot} = 0 for -3π/4 < γ < -π/2.

First excited state with finite S_{tot} has S_{tot} = 2.

Results contrast with generalized Marshall theorems for bipartite lattices.

Abstract

We consider a lattice of spin-1 particles with a general pairwise interaction $ [ {\rm cos} \gamma ({\bf S}_{l} \cdot {\bf S}_{l+1}) \ + {\rm sin} \gamma ({\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 \pi \over 4} < \gamma < - {\pi \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 $ - {\pi \over 2} < \gamma \le 0$.