Haldane-like antiferromagnetic spin chain in the large anisotropy limit

Abstract

We consider the one dimensional, periodic spin chain with $N$ sites, similar to the one studied by Haldane \cite{hal}, however in the opposite limit of very large anisotropy and small nearest neighbour, anti-ferromagnetic exchange coupling between the spins, which are of large magnitude $s$. For a chain with an even number of sites we show that actually the ground state is non degenerate and given by a superposition of the two N\'eel states, due to quantum spin tunnelling. With an odd number of sites, the N\'eel state must necessarily contain a soliton. The position of the soliton is arbitrary thus the ground state is $N$-fold degenerate. This set of states reorganizes into a band. We show that this occurs at order $2s$ in perturbation theory. The ground state is non-degenerate for integer spin, but degenerate for half-odd integer spin as is required by Kramer's theorem \cite{kram}.