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

Michael Schecter, Dimitri M. Gangardt, Alex Kamenev

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In a recent paper Gamayun et al. [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] 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); M. Schecter, D. M. Gangardt, and A. Kamanev, Ann. Phys. (N.Y.) 327, 639 (2012)]; (ii) a heavy impurity does undergo Bloch oscillations, accompanied by a drift with the velocity vD ∝ √F. 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 et al., Ann. Phys. (N.Y.) 327, 639 (2012)] in terms of Eb(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects
an interesting intermediate-force behavior.
Original languageEnglish
Article number016101
Number of pages3
JournalPhysical Review E (Statistical, Nonlinear, and Soft Matter Physics)
Issue number1
Publication statusPublished - 8 Jul 2015


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