We present a study of the dynamic structure factor of the antiferromagneticspin-1/2 Heisenberg chain at finite temperatures and finite magnetic fields.Using Quantum-Monte-Carlo based on the stochastic series expansion andMaximum-Entropy methods we evaluate the longitudinal and the transverse dynamicstructure factor from vanishing magnetic fields up to and above the threshold$B_c$ for ferromagnetic saturation, as well as for high and for intermediatetemperatures. We study the field-induced redistribution of spectral weightcontrasting longitudinal versus transverse excitations. At finite fields belowsaturation incommensurate low-energy modes are found consistent with zerotemperature Bethe-Ansatz. The crossover between the field induced ferromagnetabove $B_c$ and the Luttinger liquid below $B_c$ is analyzed in terms of thetransverse spin-dynamics. Evaluating sum-rules we assess the quality of theanalytic continuation and demonstrate excellent consistency of theMaximum-Entropy results.
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