The phase-diagram of the Blume-Capel-Haldane-Ising spin chain

TL;DR

This paper analyzes the phase diagram of a generalized one-dimensional spin chain model, identifying ground states across parameter space, discovering soliton-like excitations, and showing their size depends solely on the ratio of model parameters.

Contribution

It determines the complete ground state phase diagram of the Blume-Capel-Haldane-Ising model for arbitrary spins and reveals intrinsic soliton properties independent of system size.

Findings

Identified the ground state regions in the parameter space.

Discovered soliton-like excitations in the model.

Showed soliton size depends only on the ratio a/b.

Abstract

We consider the one-dimensional spin chain for arbitrary spin $s$ on a periodic chain with $N$ sites, the generalization of the chain that was studied by Blume and Capel \cite{bc}: $$H=\sum_{i=1}^N \left(a (S^z_i)^2+ b S^z_iS^z_{i+1}\right).$$ The Hamiltonian only involves the $z$ component of the spin thus it is essentially an Ising \cite{Ising} model. The Hamiltonian also figures exactly as the anisotropic term in the famous model studied by Haldane \cite{haldane} of the large spin Heisenberg spin chain \cite{bethe}. Therefore we call the model the Blume-Capel-Haldane-Ising model. Although the Hamiltonian is trivially diagonal, it is actually not always obvious which eigenstate is the ground state. In this paper we establish which state is the ground state for all regions of the parameter space and thus determine the phase diagram of the model. We observe the existence of solitons-like excitations and we show that the size of the solitons depends only on the ratio $a/b$ and not on the number of sites $N$. Therefore the size of the soliton is an intrinsic property of the soliton not determined by boundary conditions.