Sum rules in the dynamics of quantum spin chains

Abstract

An infinite set of sum rules is derived for the dynamics of one-dimensional quantum spin systems. They are employed to derive valuable information on the spectral-weight distribution in the $T=0$ dynamic structure factor ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q, \ensuremath{\omega})$. Applications are presented for various special cases of the nearest-neighbor $\mathrm{XXZ}$ model, including cases with a discrete excitation spectrum and cases with a continuous spectrum. For the $S=\frac{1}{2}$ $\mathrm{XY}$-Heisenberg antiferromagnet, an analytic expression for ${S}_{\mathrm{zz}}(q, \ensuremath{\omega})$ is conjectured which satisfies the infinite set of sum rules. In the $\mathrm{XY}$ limit this expression is identical to the known exact result. A similar conjecture applied to the isotropic Heisenberg antiferromagnet with arbitrary spin quantum number $S$ illustrates how the continuous spectrum of the quantum antiferromagnet collapses into a discrete branch of antiferromagnetic spin waves in the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$.