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One-dimensional Fermi polaron after a kick: two-sided singularity of the momentum distribution, Bragg reflection and other exact results
by Oleksandr Gamayun, Oleg Lychkovskiy
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Submission summary
| Authors (as registered SciPost users): | Oleg Lychkovskiy |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2404.02099v2 (pdf) |
| Date accepted: | 2024-07-09 |
| Date submitted: | 2024-06-25 10:24 |
| Submitted by: | Lychkovskiy, Oleg |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
A mobile impurity particle immersed in a quantum fluid forms a polaron - a quasiparticle consisting of the impurity and a local disturbance of the fluid around it. We ask what happens to a one-dimensional polaron after a kick, i.e. an abrupt application of a force that instantly delivers a finite impulse to the impurity. In the framework of an integrable model describing an impurity in a one-dimensional gas of fermions or hard-core bosons, we calculate the distribution of the polaron momentum established when the post-kick relaxation is over. A remarkable feature of this distribution is a two-sided power-law singularity. It emerges due to one of two processes. In the first process, the whole impulse is transferred to the polaron, without creating phonon-like excitations of the fluid. In the second process, the impulse is shared between the polaron and the center-of-mass motion of the fluid, again without creating any fluid excitations. The latter process is, in fact, a Bragg reflection at the edge of the emergent Brillouin zone. We carefully analyze the conditions for each of the two processes. The asymptotic form of the distribution in the vicinity of the singularity is derived.
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Author comments upon resubmission
We thank the Referees for their positive reports. Below we address their specific comments.
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Response to the first report.
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- quite generally a slightly more detailed discussion of the physical consequences of the results would be useful.
Some additional remarks on the experimental prospects are given in the footnote 1 on p. 9 and in the next-to-last paragraph of Sect. 4 on p. 12.
- given the recent measurements of rapidities of 1D quantum systems (see e.g. experiments by I. Bouchoule and D. Weiss) the authors could check/comment whether such measurements would bring interesting information in the case of their problem (or not) that measurements of the more macroscopic variables (such as the velocity of the impurity) would not give.
We have briefly discussed this point in the footnote 1 on p.9. In our case, expansion experiments discussed and reported in refs. [67-70] could measure the distribution of pseudomomenta $k_l$. Note, however, that in the leading order of the thermodynamic limit this distribution is simply the Fermi-Dirac distribution. All the information about the polaron is contained in the O(1/N) corrections to the Fermi-Dirac distribution, and it is unlikely that this level of precision can be attained. On the other hand, the recent experiment [71] addresses the distribution locally. It is an interesting question whether this technique can be adapted to study polarons. One immediate complication would be the a priori unknown position of the polaron ( this might be countered by repeated heralded measurements). From a more fundamental point of view, the very concept of local distribution of pseudomomenta for polaron can require clarification and justification, since the variation of the Fermi gas density around the impurity is, in general, not slow compared to interparticle distance.
- some more comments on Figure 4, in particular in connection with the two protocols given in the figure would be suitable.
A more detailed exposition of the injection protocol and its comparison to the kick protocol is given in the last paragraph of Section 3.3. on p.9
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Response to the second report.
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1- Quantify, if possible, what temperature regime future experiments might have to reach in order to at least approach the theory presented in this paper.
2- Likewise, estimate, if possible, across which timescale the experiment, or many-body numerics for that matter, would need to track the impurity dynamics in order to approach the stationary regime, i.e. when would the off-diagonal elements that have been dropped in eq. (13) would have died off sufficiently?
Roughly, the temperature must be well below the Fermi energy, while the time scale must exceed a few Fermi times, as we discuss in the next-to-last paragraph of Sect. 4 on p. 12. We plan to accurately answer these questions in a sequel to the present paper where the real-time dynamics at a finite temperature will be addressed.
3- Quantify to what extent the results shown are stable against variation of the cut-off in the many-body spectrum retained.
Firstly, let us stress that the above cut-off is not introduced as an explicit quantity and thus can not be varied in our calculations. Rather, it is introduced implicitly as follows: all eigenstates entering the sums in eqs. (13), (15) are treated as if they were from the bottom of the spectrum and thus satisfied eqs. (10), (11), (31) etc. The rationale behind such treatment is the assumption that higher-lying states with a thermodynamically large number of particle-hole excitations give a negligible contribution to these sums and thus are unimportant anyway. As the Referee correctly notes in the report, this assumption seems plausible on physical grounds but is not rigorously proven: while any individual term with O(N) particle-hole excitations is suppressed exponentially in N, as can be inferred from eq. (35), there are exponentially many such terms that, in principle, could add up to a finite contribution.
Apart from the physical intuition, the above assumption is supported by numerical tests of ref. [33,64], as we point out in the revised version (see a sentence below eq. (13)). The final resolution of this issue will be presented in the above-mentioned sequel to the present paper, where we will abandon the above assumption and rigorously account for finite-entropy states.
List of changes
1. A sentence added below eq. (13). It clarifies the grounds for considering only low-lying states.
2. Footnote 1 added.
3. A paragraph on the injection protocol added in the end of Sect. 3.3.
4. A paragraph discussing rough estimates of relevant temperature and time scales added in the end of Sect. 4.
5. Minor style and grammar improvements introduced.
6. Refs. [64,67-71] introduced.
Published as SciPost Phys. 17, 063 (2024)