irrelevant in the linear spin wave approximation. Such a result can naturally be expected following basic quantum mechanics but would not be achieved in the well-known double exchange picture [@Zener1951], for in that classical approach the hole would not be able to hop at all in the $S=1$ antiferromagnet.

Acknowledgments. {#acknowledgments. .unnumbered}
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We are very grateful to David Sutter and Johan Chang for sharing the experimental data that was presented in Ref.  and which allowed us to plot Fig. \[result\](a). We thank Eugenio Paris for insightful discussions. A.K. thanks the IFW Dresden for the kind hospitality. A.K. and K.W. (K.W.) acknowledge(s) support by Narodowe Centrum Nauki (NCN, Poland) under Projects No. 2016/22/E/ST3/00560 (2016/23/B/ST3/00839), respectively. J.v.d.B. acknowledges financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) via SFB1143 project A5 and the Würzburg-Dresden Cluster of Excellence ct.qmat.

APPENDIX: Derivation of the Polaron model {#sec:appendix .unnumbered}
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Mapping onto a polaronic model: Spin Hamiltonian ($\mathcal{H}_J$)
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To diagonalize the spin $S=1$ Hamiltonian discussed in the paper \[cf. Eq. (\[model1\]) of the main text of the paper\] we start by performing two rotations of spins. First, we make a different choice of the spin quantization axis $\hat{z}$ - in this case 