(Abridged) Debris disks trace remnant reservoirs of leftover planetesimals in planetary systems. A handful of warm debris disks have been discovered in the last years, where emission in excess starts in the mid-infrared. An interesting subset within these warm debris disks are those where emission features are detected in mid-IR spectra, which points towards the presence of warm micron-sized dust grains. Given the ages of the host stars, the presence of these grains is puzzling, and questions their origin and survival in time. This study focuses on determining the mineralogy of the dust around 7 debris disks with evidence for warm dust, based on Spitzer/IRS spectroscopic data, in order to provide new insights into the origin of the dust grains. We present a new radiative transfer code dedicated to SED modeling of optically thin disks. We make use of this code on the SEDs of seven warm debris disks, in combination with recent laboratory experiments on dust optical properties. We find that most, if not all, debris disks in our sample are experiencing a transient phase, suggesting a production of small dust grains on relatively short timescales. From a mineralogical point of view, we find that enstatite grains have small abundances compared to crystalline olivine grains. The main result of our study is that we find evidences for Fe-rich crystalline olivine grains (Fe / [Mg + Fe] ~ 0.2) for several debris disks. This finding contrasts with studies of gas-rich protoplanetary disks. The presence of Fe-rich olivine grains, and the overall differences between the mineralogy of dust in Class II disks compared to debris disks suggest that the transient crystalline dust is of a new generation. We discuss possible crystallization routes to explain our results, and comment on the mechanisms that may be responsible for the production of small dust grains.
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