Spin-Wave Spectrum of the Antiferromagnetic Linear Chain

Abstract

The methods of Bethe and Hulth\'en 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 $H=\ensuremath{\Sigma}{j}^{}{\mathrm{S}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\ensuremath{\epsilon}}_{k}=(\frac{\ensuremath{\pi}}{2})|sink|$, whereas Anderson's spin-wave theory gives ${\ensuremath{\epsilon}}_{k}=|sink|$. 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.