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Comment on "Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas"

In a recent paper, arxiv:1402.6362, Gamayun, Lychkovskiy, and Cheianov studied the dynamics of a mobile impurity embedded into a one-dimensional Tonks-Girardeau gas of strongly interacting bosons. Employing the Boltzmann equation approach, they arrived at the following main conclusions: (i) a light impurity, being accelerated by a constant force does not exhibit Bloch oscillations; (ii) a heavy impurity does undergo Bloch oscillations, accompanied by a drift with the velocity proportional to the square root of force. In this comment we argue that the result (i) is an artifact of the classical Boltzmann approximation, which misses the formation of the (quasi) bound-state between the impurity and a hole. Result (ii), while not valid at asymptotically small force, indeed reflects an interesting intermediate-force behavior. Here we clarify its limits of applicability and extend beyond the Tonks-Girardeau limit.