Fundamental conservation laws predict ballistic, i.e., dissipationless transport behaviour in one-dimensional quantum magnets. Experimental evidence, however, for such anomalous transport has been lacking ever since. Here we provide experimental evidence for ballistic heat transport in a S=1/2 Heisenberg chain. In particular, we investigate high purity samples of the chain cuprate SrCuO2 and observe a huge magnetic heat conductivity $kappa_{mag}$. An extremely large spinon mean free path of more than a micrometer demonstrates that $kappa_{mag}$ is only limited by extrinsic scattering processes which is a clear signature of ballistic transport in the underlying spin model.
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