Dynamic correlations of antiferromagnetic spin- XXZ chains at arbitrary temperature from complete diagonalization

Abstract

All eigenstates and eigenvalues are determined for the spin- XXZ chain H=2J${\ensuremath{\sum}}_{\mathrm{i}}$(${\mathrm{S}}_{\mathrm{i}}^{\mathrm{x}}$${\mathrm{S}}_{\mathrm{i}+1}^{\mathrm{x}}$+${\mathrm{S}}_{\mathrm{i}}^{\mathrm{y}}$${\mathrm{S}}_{\mathrm{i}+1}^{\mathrm{y}}$+\ensuremath{\Delta}${\mathrm{S}}_{\mathrm{i}}^{\mathrm{z}}$${\mathrm{S}}_{\mathrm{i}+1}^{\mathrm{z}}$) for rings with up to N=16 spins, for anisotropies \ensuremath{\Delta}=0, cos (0.3\ensuremath{\pi}), and 1. The dynamic spin pair correlations 〈${\mathrm{S}}_{\mathrm{l}+\mathrm{n}}^{\mathrm{\ensuremath{\mu}}}$(t)${\mathrm{S}}_{\mathrm{l}}^{\mathrm{\ensuremath{\mu}}}$〉, (\ensuremath{\mu}=x,z), the dynamic structure factors ${\mathrm{S}}^{\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), and the intermediate structure factors ${\mathrm{I}}^{\mathrm{\ensuremath{\mu}}}$(q,t) are calculated for arbitrary temperature T. It is found that for all T, ${\mathrm{S}}^{\mathrm{z}}$(q,\ensuremath{\omega}) is mainly concentrated on the region |\ensuremath{\omega}|${\mathrm{\ensuremath{\varepsilon}}}_{2}$(q), where ${\mathrm{\ensuremath{\varepsilon}}}_{2}$(q) is the upper boundary of the two-spinon continuum, although excited states corresponding to a much broader frequency spectrum contribute. This is also true for the Haldane-Shastry model and the frustrated Heisenberg model. The intermediate structure factors ${\mathrm{I}}^{\mathrm{\ensuremath{\mu}}}$(q,t) for \ensuremath{\Delta}\ensuremath{\ne}0 show exponential decay for high T and large q. Within the accessible time range, the time-dependent spin-correlation functions do not display the long-time signatures of spin diffusion.