# Beyond Gross-Pitaevskii equation for 1D gas: quasiparticles and solitons

### Submission summary

 As Contributors: Jakub Kopyciński · Maciej Łebek Arxiv Link: https://arxiv.org/abs/2106.15289v3 (pdf) Date accepted: 2021-11-24 Date submitted: 2021-11-03 14:30 Submitted by: Kopyciński, Jakub Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory

### Abstract

Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.

Published as SciPost Phys. 12, 023 (2022)

### List of changes

1. Abstract improvement.
2. Extension of the reference list.
3. Clarification of the text in Sec. 4.2.
4. Addition of Figs. 3, 8-11.
5. Correction of minor typos and issues.

### Submission & Refereeing History

Resubmission 2106.15289v3 on 3 November 2021
Submission 2106.15289v2 on 20 July 2021

## Reports on this Submission

### Anonymous Report 2 on 2021-11-19 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2106.15289v3, delivered 2021-11-19, doi: 10.21468/SciPost.Report.3875

### Report

See my attached report.

### Attachment

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Anonymous Report 1 on 2021-11-7 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2106.15289v3, delivered 2021-11-07, doi: 10.21468/SciPost.Report.3805

### Report

The authors have carefully answered all my comments and amended their manuscript correspondingly.

I am happy to recommend publication in SciPost Physics once the authors have considered a few remaining/newly arising questions:

1. Once again to the formulation that "..the LLGPE soliton is close to G_2 of the fermionized gas is alarming in the context of the hydrodynamical origin.." at the end of Sect. 4.

I now see the proximity of the G_2 and the soliton solution, what I do not understand, yet, though, is why the authors consider this an "alarming" observation. Do they mean "interesting" or "striking"? Otherwise, if one should worry about this proximity, it would be useful to have an explanation. I would expect that in the Tonks limit, a soliton should not have a spatial profile larger than the interaction radius of the atoms. As the fermionization tells, particles no longer want to come close to each other, so that explains the shape of the G_m for any m, as the authors point out. But shouldn't a soliton in the Tonks limit be something like a fermionic hole excitations, possibly arising out of a pair formation process? In that respect I would not consider this finding as alarming but as consistent with the expectation, or, thinking closer about it, as interesting, given the fact that type I and type II branch particles at a particular momentum are not really separate excitations in the TG limit (taking into account the findings of Brand and Cherny again)? Maybe the authors could provide still a few clarifying remarks?

2. A technical question in this context: Where is x' in the top right panel of Fig. 7? It is indicated in the TG limit in the lower right panel, but not in the NI limit. Do I have to consider, here, x' to be given by the shown distribution of G_2?

Further to the NI limit: I understand that there should be soliton-like solutions in the ideal gas, if there are such in the WI limit of any small gamma. But shouldn't such solutions correspond to product states of single-particle wave functions showing a dip at a particular point. In their cited paper PRA97, 06317 (2018), the authors derive such shapes which just look like that: cosine waves in the periodic boundary conditions which have a dip in the center of the volume. I would imagine a product of such free wave functions to form something like a dark soliton. And as I take from that paper, these "solitons" are in fact dispersive. So should one really talk of "solitons" in that limit? I find this a bit misleading and at least requiring explanations.

3. A few more minor points (typos etc):

- Intro, 1st sentence: In weakly....Bose gases, ...
- p. 4, 4th last line: "find to be useful" -> "are found to be useful"?
- p. 7, par. before Eq. 18: occurs in ... Bose gases
- p. 12, 3rd last line: "in the log.." -> "on a double-log..."
- p. 13, 2nd last line of 1st par, Sect. 4.2: static: stationary?
- p. 14, 1st line: to infinity
- p. 14, 4th last par., 2nd line: authors
- p. 16, Sect. 6, 4th l.: giving stronger foundations for further extensions, like...?
- dois missing in Refs. 9,10,17-20,25

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Jakub Kopyciński  on 2021-11-29  [id 1991]

(in reply to Report 1 on 2021-11-07)

Dear Referee,

1.

Once again to the formulation that the LLGPE soliton is close to $G_2$ of the fermionized gas is alarming in the context of the hydrodynamical origin at the end of Sect. 4. I now see the proximity of the $G_2$ and the soliton solution, what I do not understand, yet, though, is why the authors consider this an "alarming" observation. Do they mean "interesting" or "striking"? Otherwise, if one should worry about this proximity, it would be useful to have an explanation. I would expect that in the Tonks limit, a soliton should not have a spatial profile larger than the interaction radius of the atoms. As the fermionization tells, particles no longer want to come close to each other, so that explains the shape of the $G_m$ for any $m$, as the authors point out. But shouldn't a soliton in the Tonks limit be something like a fermionic hole excitations, possibly arising out of a pair formation process? In that respect I would not consider this finding as alarming but as consistent with the expectation, or, thinking closer about it, as interesting, given the fact that type I and type II branch particles at a particular momentum are not really separate excitations in the TG limit (taking into account the findings of Brand and Cherny again)? Maybe the authors could provide still a few clarifying remarks?

Finally, we change "alarming" to "striking".

Our problem with this overlap between solitons and $G_2$ function in the TG regime is twofold: a) the relation between the yrast state and the soliton is completely different that to what we used to in the weakly interacting regime. For weak interaction, the yrast state is close to a continuous superposition of mean-field solitons, with a random position. It doesn't seem the case in the TG regime. b) As written in the text, this coincidence happens beyond the hydrodynamic equation, namely, it is impossible to define "fluid elements" consisting of many atoms building the solitonic profile.

But indeed, in the LL quasimomenta picture, a soliton is like particle-hole excitation. Also, in the TG regime, one can easily construct type I excitations from type II ones.

2.

A technical question in this context: Where is $x'$ in the top right panel of Fig. 7? It is indicated in the TG limit in the lower right panel, but not in the NI limit. Do I have to consider, here, $x'$ to be given by the shown distribution of $G_2$?

We replot the figure to make everything clear. We indicate $x'$ in both panels. Indeed $x'$ needs to be given by the shown distribution of $G_2$.

Further to the NI limit: I understand that there should be soliton-like solutions in the ideal gas, if there are such in the WI limit of any small gamma. But shouldn't such solutions correspond to product states of single-particle wave functions showing a dip at a particular point. In their cited paper PRA97, 06317 (2018), the authors derive such shapes which just look like that: cosine waves in the periodic boundary conditions which have a dip in the center of the volume. I would imagine a product of such free wave functions to form something like a dark soliton. And as I take from that paper, these "solitons" are in fact dispersive. So should one really talk of "solitons" in that limit? I find this a bit misleading and at least requiring explanations.

Strictly speaking the type II excitations in the NI limit are highly entangled (due to symmetrization) states. Hence they cannot be reduced to simple product states. But once the symmetry is broken, the system may fall into a product state, in which all particles occupy a density dip. After this symmetry breaking procedure, the dip moves with a constant velocity, preserving its shape.

In our paper we have also shown that the yrast states corresponding to gray solitons can be considered as composed of mean-field solitons, but with different momenta. In each symmetry breaking event one find a mean-field soliton, but with a random depth. 3.

A few more minor points (typos etc): - Intro, 1st sentence: In weakly....Bose gases, ... - p. 4, 4th last line: "find to be useful" -> "are found to be useful"? - p. 7, par. before Eq. 18: occurs in ... Bose gases - p. 12, 3rd last line: "in the log.." -> "on a double-log..." - p. 13, 2nd last line of 1st par, Sect. 4.2: static: stationary? - p. 14, 1st line: to infinity - p. 14, 4th last par., 2nd line: authors - p. 16, Sect. 6, 4th l.: giving stronger foundations for further extensions, like...? - dois missing in Refs. 9,10,17-20,25

These minor changes will be also included before the publication.

Thank you once again for the careful reading of our manuscript, Jakub Kopyciński Maciej Łebek Maciej Marciniak Wojciech Górecki Rafał Ołdziejewski Krzysztof Pawłowski