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

In a recent paper Gamayun et al. [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)PLEEE81539-375510.1103/PhysRevE.90.032132] studied the dynamics of a mobile impurity weakly coupled to a one-dimensional Tonks-Girardeau gas of strongly interacting bosons. Employing the Boltzmann equation approach, they, in particular, arrived at the following conclusions: (i) a light impurity, being accelerated by a constant force F, does not exhibit Bloch oscillations, which were predicted and studied by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.070402; M. Schecter, D. M. Gangardt, and A. Kamanev, Ann. Phys. (N.Y.) 327, 639 (2012)APNYA60003-491610.1016/j.aop.2011.10.001]; (ii) a heavy impurity does undergo Bloch oscillations, accompanied by a drift with the velocity vDF. In this Comment we argue that result (i) is an artifact of the classical Boltzmann approximation. The latter misses the formation of the quasibound state between the impurity and a hole. Its dispersion relation Eb(P,ρ) is a smooth periodic function of momentum P with the period 2kF=2πρ, where ρ is a density of the host gas. Being accelerated by a small force, such a bound-state exhibits Bloch oscillations superimposed with the drift velocity vD=μF. The mobility μ may be expressed exactly [M. Schecter, Ann.Phys. (N.Y.) 327, 639 (2012)APNYA60003-491610.1016/j.aop.2011.10.001] in terms of Eb(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects an interesting intermediate-force behavior.