Spin-Wave Spectrum of the Antiferromagnetic Linear Chain

Abstract

The methods of Bethe and Hulthén are used to build spin-wave states for the antiferromagnetic linear chain. These states, of spin 1 and translational quantum number $k$, are eigenstates of the Hamiltonian $H=\Sigma j^{}{\mathrm{S}}_j·{\mathrm{S}}_{j+1\mathrm{}}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\epsilon }_k=\left(\frac{\pi }{2}\right)\left|sink\right|$, whereas Anderson's spin-wave theory gives ${\epsilon }_k=\left|sink\right|$. For finite chains it has been verified by numerical computation that these states are the lowest states of given $k$, but no rigorous proof has been given for an infinite chain.