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The heat transport in a system of $S=1/2$ large-$J$ Heisenberg spin chains,describing closely Sr$_2$CuO$_3$ and SrCuO$_2$ cuprates, is studiedtheoretically at $T\ll J$ by considering interactions of the bosonized spinexcitations with optical phonons and defects. Treating rigorously themulti-boson processes, we derive a microscopic spin-phonon scattering rate thatadheres to an intuitive picture of phonons acting as thermally populateddefects for the fast spin excitations. The mean-free path of the latterexhibits a distinctive $T$-dependence reflecting a critical nature of spinchains and gives a close description of experiments. By the naturalnesscriterion of realistically small spin-phonon interaction, our approach standsout from previous considerations that require large coupling constants toexplain the data and thus imply a spin-Peierls transition, absent in realmaterials.

Heat Transport in Spin Chains with Weak Spin-Phonon Coupling