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$\mathrm{T}\overline{\mathrm{T}}$-deformed 1d Bose gas
by Yunfeng Jiang
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Submission summary
| Authors (as registered SciPost users): | Yunfeng Jiang |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2011.00637v2 (pdf) |
| Date submitted: | 2021-12-20 04:33 |
| Submitted by: | Jiang, Yunfeng |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
$\mathrm{T}\overline{\mathrm{T}}$ deformation was originally proposed as an irrelevant solvable deformation for 2d relativistic quantum field theories (QFTs). The same family of deformations can also be defined for integrable quantum spin chains which was first studied in the context of integrability in AdS/CFT. In this paper, we construct such deformations for yet another type of models, which describe a collection of particles moving in 1d and interacting in an integrable manner. The prototype of such models is the Lieb-Liniger model. This shows that such deformations can be defined for a very wide range of systems. We study the finite volume spectrum and thermodynamics of the $\mathrm{T}\overline{\mathrm{T}}$-deformed Lieb-Liniger model. We find that for one sign of the deformation parameter $(\lambda<0)$, the deformed spectrum becomes complex when the volume of the system is smaller than certain critical value, signifying the break down of UV physics. For the other sign $(\lambda>0)$, there exists an upper bound for the temperature, similar to the Hagedorn behavior of the $\mathrm{T}\overline{\mathrm{T}}$ deformed QFTs. Both behaviors can be attributed to the fact that $\mathrm{T}\overline{\mathrm{T}}$ deformation changes the size the particles. We show that for $\lambda>0$, the deformation increases the spaces between particles which effectively increases the volume of the system. For $\lambda<0$, $\mathrm{T}\overline{\mathrm{T}}$ deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of $\mathrm{T}\overline{\mathrm{T}}$-deformed free boson is the Nambu-Goto action, which describes bosonic strings -- also an extended object with finite size.
Author comments upon resubmission
Reply to Referee 1.
I would like to thank the referee for the careful work. I made changes according to the suggestion of the referee. These are given in the List of changes.
Reply to Referee 2.
I would like to thank the referee for the careful and very helpful work. Here is the answer to the points raised by the referee.
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I agree that neglecting the term $\mathcal{Y}_{JJ}$ contain some subtlety as pointed out by the referee. Therefore the fact that we can neglect such a term depends on the assumption that currents vanish or negligible at infinity. On the other hand, we do expect that such assumptions should hold for rather general situations. The point is that on a infinite line the bilocal/bilinear deformation do not change the spectrum of the model. This is consistent with the fact of neglecting the $\mathcal{Y}_{JJ}$ term.
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I thank the referee for raising this point. Indeed it was not clarified in the draft that in fact the partially ordered state $|u<v\rangle$ should be understood as an \emph{asymptotic state}, namely the two particles are far way from each other. Notice that the S-matrix is defined by the ratio of the coefficients in front of the asymptotic states and is not sensitive to the ``UV'' details concerning when the two particles are close to each other. Therefore, (38) should be understood as an approximate eigenstate instead of an exact Bethe state of the deformed Hamiltonian. For such asymptotic states, it is easy to see that our derivation holds. Since the particles are widely separated, they can be seen approximately as a particle moving on an infinite line, so that sweeping the density operator over the corresponding region gives the corresponding charge of that particle. We modified (39) and made a few more comments to clarify the point.
Other modifications are listed in the List of changes.
List of changes
List of changes (according to the comments of referee 1)
1. The typo above Eq(49) has been corrected;
2. I have corrected the punctuation at the end of equations (22),(49),(80),(112),(127),(128) together with a few more other equations.
List of changes (according to the comments of referee 2)
1. (38) and (39) are modified. Few comments are added around these equations.
2. I thank the referee for this comment. I added this comment in the sentence below eq(50).
3. On page 15, I make a comment on the relation to GHD and cited the papers suggested by the referee.
4. On page 25, I added a short sentence to mention that the generalized current operator was first introduced in reference [g] proposed by the referee.
5. On page 26, I added that the effective velocity was introduced in [a][b] and first appeared in [h].
6. The sentence at the bottom of page 9 is now corrected.
7. Period added after (9).
8. Typo corrected after (65).
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2022-2-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2011.00637v2, delivered 2022-02-07, doi: 10.21468/SciPost.Report.4335
Report
The author has answered my questions well. There is just one point on which I think the author should comment quickly in the manuscript: the vanishing of the Y term in eq 27 holds under the assumption that all currents vanish at infinity, and this is true if, for instance, one finds the vacuum at infinity. This is fine for asymptotic states, which is what the author considers here, but I think it is important to clarify this point, as it forbids for instance the large-volume limit of finite-density states. When this is done then the paper should be published.
Requested changes
Small comment about negleting Y term after eq 27.
Report #1 by Anonymous (Referee 3) on 2021-12-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2011.00637v2, delivered 2021-12-23, doi: 10.21468/SciPost.Report.4085
Strengths
1- It is essentially the first paper discussing the TTbar deformation of the Lieb-Liniger model in such details.
2- The background sections are carefully written and very useful.
3- The topic is interesting and important.
4- The paper contains many original analytic and numerical results.
Weaknesses
- no evident weaknesses to signal
Report
The author amended the paper according to the comments of referees 1 and 2, now it can be published without further delay.