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Physical Review A

Reply to “Comment on ‘Motion of an impurity particle in an ultracold quasi-one-dimensional gas of hard-core bosons’ ”

Marvin D. Girardeau, Anna Minguzzi

Published Sep 7, 2010• 2 citations
Publisher

Abstract

In their Comment Giraud and Combescot (GC) [Phys. Rev. A 82, 037603 (2010)] point out that the contribution to the impurity-boson distribution function ${\ensuremath{\rho}}_{\mathit{bi}}(x\ensuremath{-}y)$ of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but our results for ${\ensuremath{\rho}}_{\mathit{bi}}$ are highly accurate for large impurity-boson mass ratio ${m}_{i}/m$, for all values of $x\ensuremath{-}y$ and boson-impurity coupling constant.

DOI
10.1103/physreva.82.037604

Related papers

Reply to arXiv:1002.4366, "Comment on `Motion of an impurity in an ultracold quasi-one-dimensional gas of hard-core bosons`", by S. Giraud and R. Combescot

Marvin D. Girardeau, Anna Minguzzi

In their Comment [1] Giraud and Combescot point out that the contribution to the impurity-boson distribution function $ρ_{bi}(x-y)$ of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but nevertheless our results for $ρ_{bi}$ are highly accurate for large impurity-boson mass ratio $m_i/m$ and remain qualitatively correct for all values of $m_i/m$ and all values of the boson-impurity coupling constant.

arXiv:1003.0652•Mar 2, 2010•1 citation

Comment on “Motion of an impurity particle in an ultracold quasi-one-dimensional gas of hard-core bosons”

S. Giraud, R. Combescot

Very recently Girardeau and Minguzzi [Phys. Rev. A 79, 033610 (2009)] have studied an impurity in a one-dimensional gas of hard-core bosons. In particular they dealt with the general case where the mass of the impurity is different from the mass of the bosons and the impurity-boson interaction is not necessarily infinitely repulsive. We show that one of their initial steps is unjustified, contradicting known exact results. Their results in the general case apply only actually when the mass of the impurity is infinite.

arXiv:1002.4366•Physical Review A•Sep 7, 2010•2 citations

Motion of an impurity particle in an ultracold quasi-one-dimensional gas of hard-core bosons

Marvin D. Girardeau, Anna Minguzzi

The low-lying energy eigenstates of a one-dimensional (1D) system of many impenetrable point bosons and one moving impurity particle with repulsive zero-range impurity-boson interaction are found for all values of the impurity-boson mass ratio and coupling constant. The moving entity is a polaronlike composite object consisting of the impurity clothed by a comoving gray soliton. The special case with impurity-boson interaction of point hard-core form and impurity-boson mass ratio ${m}_{i}/m$ unity is first solved exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given using second quantization for the bosons and the second-quantized form of the FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines canonical transformation. This yields the exact ground state (total linear momentum $q=0$) and exact boson-impurity distribution function in the thermodynamic limit for arbitrary ${m}_{i}/m$ and arbitrary impurity-boson interaction strength. These results are then extended to states with $q>0$.

arXiv:0807.3366•Physical Review A•Mar 12, 2009•23 citations

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

Oleksandr Gamayun, Oleg Valentinovich Lychkovskiy, Vadim Cheianov

In our recent paper [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] we studied the dynamics of a mobile impurity particle weakly interacting with the Tonks-Girardeau gas and pulled by a small external force F. Working in the regime when the thermodynamic limit is taken prior to the small force limit, we found that the Bloch oscillations of the impurity velocity are absent in the case of a light impurity. Further, we argued that for a light impurity the steady-state drift velocity V(D) remains finite in the limit F→0. These results are in contradiction with earlier works by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamenev, Ann. Phys. (NY) 327, 639 (2012)]. One of us has conjectured [O. Lychkovskiy, Phys. Rev. A 91, 040101 (2015)] that the central assumption of these works, the adiabaticity of the dynamics, can break down in the thermodynamic limit. In the preceding Comment [M. Schecter, D. M. Gangardt, and A. Kamenev, Phys. Rev. E 92, 016101 (2015)], Schecter et al. have argued against this conjecture and in support of the existence of Bloch oscillations and the linearity of V(D)(F). They have suggested that the ground state of the impurity-fluid system is a quasibound state and that this is sufficient to ensure adiabaticity in the thermodynamic limit. Their analytical argument is based on a certain truncation of the Hilbert space of the system. We argue that extending the results and intuition based on their truncated model on the original many-body problem lacks justification.

arXiv:1407.4433•Physical Review E•Jul 8, 2015•11 citations

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

Michael Schecter, Dimitri M. Gangardt, Alex Kamenev

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 v(D)∝√[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 E(b)(P,ρ) is a smooth periodic function of momentum P with the period 2k(F)=2πℏρ, 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 v(D)=μF. The mobility μ may be expressed exactly [M. Schecter et al., Ann. Phys. (N.Y.) 327, 639 (2012)] in terms of E(b)(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects an interesting intermediate-force behavior.

arXiv:1404.4366•Physical Review E•Jul 8, 2015•14 citations

Reply to “Comment on ‘Accurate ground-state phase diagram of the one-dimensional extended Hubbard model at half filling’ ”

G. P. Zhang

We respond to Jeckelman's Comment on our paper [G. P. Zhang, Phys. Rev. B 68, 153101 (2003)], which discusses his previous work [E. Jeckelmann, Phys. Rev. Lett. 89, 236401 (2002)]. We feel that it is important to point out that his previous results show a trend contradicting the weak-coupling limit, a conclusion that is independent of our calculation. Therefore, our comments on his previous results remain valid and our conclusion is unchanged.

Physical Review B•May 4, 2005•2 citations
In their Comment Giraud and Combescot (GC) [Phys. Rev. A 82, 037603 (2010)] point out that the contribution to the impurity-boson distribution function \ensuremathρbi(x\ensuremath−y){\ensuremath{\rho}}_{\mathit{bi}}(x\ensuremath{-}y)\ensuremathρbi​(x\ensuremath−y) of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but our results for \ensuremathρbi{\ensuremath{\rho}}_{\mathit{bi}}\ensuremathρbi​ are highly accurate for large impurity-boson mass ratio mi/m{m}_{i}/mmi​/m, for all values of x\ensuremath−yx\ensuremath{-}yx\ensuremath−y and boson-impurity coupling constant.
In their Comment [1] Giraud and Combescot point out that the contribution to the impurity-boson distribution function ρbi(x−y)ρ_{bi}(x-y)ρbi​(x−y) of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but nevertheless our results for ρbiρ_{bi}ρbi​ are highly accurate for large impurity-boson mass ratio mi/mm_i/mmi​/m and remain qualitatively correct for all values of mi/mm_i/mmi​/m and all values of the boson-impurity coupling constant.
Very recently Girardeau and Minguzzi [Phys. Rev. A 79, 033610 (2009)] have studied an impurity in a one-dimensional gas of hard-core bosons. In particular they dealt with the general case where the mass of the impurity is different from the mass of the bosons and the impurity-boson interaction is not necessarily infinitely repulsive. We show that one of their initial steps is unjustified, contradicting known exact results. Their results in the general case apply only actually when the mass of the impurity is infinite.
The low-lying energy eigenstates of a one-dimensional (1D) system of many impenetrable point bosons and one moving impurity particle with repulsive zero-range impurity-boson interaction are found for all values of the impurity-boson mass ratio and coupling constant. The moving entity is a polaronlike composite object consisting of the impurity clothed by a comoving gray soliton. The special case with impurity-boson interaction of point hard-core form and impurity-boson mass ratio mi/m{m}_{i}/mmi​/m unity is first solved exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given using second quantization for the bosons and the second-quantized form of the FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines canonical transformation. This yields the exact ground state (total linear momentum q=0q=0q=0) and exact boson-impurity distribution function in the thermodynamic limit for arbitrary mi/m{m}_{i}/mmi​/m and arbitrary impurity-boson interaction strength. These results are then extended to states with q>0q>0q>0.
In our recent paper [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] we studied the dynamics of a mobile impurity particle weakly interacting with the Tonks-Girardeau gas and pulled by a small external force F. Working in the regime when the thermodynamic limit is taken prior to the small force limit, we found that the Bloch oscillations of the impurity velocity are absent in the case of a light impurity. Further, we argued that for a light impurity the steady-state drift velocity V(D) remains finite in the limit F→0. These results are in contradiction with earlier works by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamenev, Ann. Phys. (NY) 327, 639 (2012)]. One of us has conjectured [O. Lychkovskiy, Phys. Rev. A 91, 040101 (2015)] that the central assumption of these works, the adiabaticity of the dynamics, can break down in the thermodynamic limit. In the preceding Comment [M. Schecter, D. M. Gangardt, and A. Kamenev, Phys. Rev. E 92, 016101 (2015)], Schecter et al. have argued against this conjecture and in support of the existence of Bloch oscillations and the linearity of V(D)(F). They have suggested that the ground state of the impurity-fluid system is a quasibound state and that this is sufficient to ensure adiabaticity in the thermodynamic limit. Their analytical argument is based on a certain truncation of the Hilbert space of the system. We argue that extending the results and intuition based on their truncated model on the original many-body problem lacks justification.
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 v(D)∝√[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 E(b)(P,ρ) is a smooth periodic function of momentum P with the period 2k(F)=2πℏρ, 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 v(D)=μF. The mobility μ may be expressed exactly [M. Schecter et al., Ann. Phys. (N.Y.) 327, 639 (2012)] in terms of E(b)(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects an interesting intermediate-force behavior.