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Emptiness Instanton in Quantum Polytropic Gas
by Alexander G. Abanov, Dimitri M. Gangardt
Submission summary
| Authors (as registered SciPost users): | Dimitri Gangardt |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2412.11686v3 (pdf) |
| Date submitted: | 2025-03-06 16:51 |
| Submitted by: | Gangardt, Dimitri |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The emptiness formation problem is addressed for a one-dimensional quantum polytropic gas characterized by an arbitrary polytropic index $\gamma$, which defines the equation of state $P \sim \rho^\gamma$, where $P$ is the pressure and $\rho$ is the density. The problem involves determining the probability of the spontaneous formation of an empty interval in the ground state of the gas. In the limit of a macroscopically large interval, this probability is dominated by an instanton configuration. By solving the hydrodynamic equations in imaginary time, we derive the analytic form of the emptiness instanton. This solution is expressed as an integral representation analogous to those used for correlation functions in Conformal Field Theory. Prominent features of the spatiotemporal profile of the instanton are obtained directly from this representation.
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- Provide a novel and synergetic link between different research areas.
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Dear Editor,
We are grateful to you and the Referees for reviewing our manuscript and recommending its publication in SciPost. We have followed the Referees' comments and suggestions to prepare the amended version, which we are resubmitting after some minor corrections. Below are our responses to the Referees' comments:
Referee 1
1- the sentence above eq.(27) mentions "in the form of series", while eq.(27) itself looks to me not a series, but a finite sum. Should it be "in the form of sums" instead?
-- We have changed the sentence referring to the Eq. 27 as "the sum".
2- an apparent typo in the caption to the left panel of Fig.3. It mentions Eqs. (31) and (33), while it should be (32) and (33), to my understanding.
-- We have corrected the equation numbers.
3- in the footnote in p.3, as well as in the caption to Fig.2, the authors mention the Chaplygin gas with n=-1 (gamma=-1). So the reader is tempted to think that the results of the paper may somehow be analytically continued to that case, too. However, both in the figure caption and after Eq.(11) the authors specify that the analysis of the paper is only valid for gamma>1 (and thus is not valid for the Chaplygin gas). It would be helpful to the reader to clarify (possibly in the last section of the paper) if there are any implications for Chaplygin gas or for any other values gamma<1 or not. Otherwise mentioning this case and an analytic continuation in Fig.2 looks confusing.
-- We have addressed the Referee's concern about the validity range of our results and have added a sentence at the end of the first paragraph of the 'Conclusions and Open Questions' section as recommended. We believe that this way presenting some of the parameters of the Emptiness Instanton solution is justified outside the validity range, and this is clearly explained.
Referee 2 has not requests
Referee 3
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Page 7, Eq (28): the last factor in numerator, (n−m+1) , should be corrected into (n+m−1);
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page 10, sentence before Eq (47): asymptitics should be corrected into asymptotic;
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page 16, 1st line after Eq (60): the inline equation w±=u±c should be corrected into w±=v±c.
-- All corrected
We believe that the current version can be published.
Sincerely yours,
AG Abanov and DM Gangardt
List of changes
1. The sentence above Eq 27 is changed to refer to 'the sum' instead of 'the series'.
2. Equation numbers in the caption to Fig. 3 are corrected.
3. A sentence 'While the solution for the Emptiness Instanton.. ' is added at the end of the first paragraph of Section 6.
4. The factor in Eq (28) is corrected.
5. Spelling of 'asymptotics' is corrected before Eq. (47).
6. The inline equation after Eq. (6) is corrected.
7. We have modified the sentence after Eq. (1) which now contains integer values of n relevant to one-dimensional gases.
8. We added Ilya Gruzberg and Paul Krapivsky in the acknowledgments.