SciPost Submission Page
Emptiness Instanton in Quantum Polytropic Gas
by Alexander G. Abanov, Dimitri M. Gangardt
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Dimitri Gangardt |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2412.11686v2 (pdf) |
| Date submitted: | 2024-12-30 14:59 |
| 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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
The result are original, and solve a nontrivial problem.
The paper is well written, and very instructive.
Weaknesses
no weaknesses
Report
The paper addresses the problem of emptiness formation in a one-dimensional polytropic gas (defined by the equation of state $P\sim\rho^\gamma$), by using a large-deviation (or instanton) calculation, and solving the related hydrodynamic equations, along the line of [24]. The paper provides an exact analytic solution to the problem for generic values of $\gamma>1$. The calculations and results extends previous ones obtained by one of the authors [5] for the discrete set of values $\gamma=1/(2n+1)$, with $n$ positive integer. The physical implications of the obtained result are discussed. The technical aspects of the derivation, although far from trivial, are explained in a very clear and instructive way. The whole paper is well and clearly written.
In conclusion, the paper fulfills completely all necessary requirements of riginality, scientific rigour, relevance, clarity, and interest.
I strongly recommend publication, modulo corrections of a few typos, see below.
Requested changes
- Page 7, Eq (28): the last factor in numerator, $(n-m+1)$, should be corrected into $(n+m-1)$;
- page 10, sentence before Eq (47): asymptitics should be corrected into
asymptotics;
"- page 16, 1st line after Eq (60): the inline equation $w_{\pm}=u\pm c$ should be corrected into $w_{\pm}=v\pm c$.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1. The results are original and to some may appeal as elegant.
2. The paper is well written and easy to read.
Weaknesses
1. The results represent an incremental, if aesthetically satisfying, improvement on one of the authors' previous work.
Report
The manuscript presents an extension of one of the authors' earlier result regarding the semiclassical emptiness formation probability in a one-dimensional Galilean quantum fluid. The main result of the paper is the
proof of the formula for the emptiness formation probability in the case
of a non-integer polytropic index.
The paper is well written and is mathematically sound.
The techniques used to derive the main result may be found instructive for researchers interested in instanton calculus and one-dimensional physics.
I believe that the paper easily meets the criteria of SciPost and deserves publication without change.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
This paper is a follow-up to the earlier work Ref.[5]. The main result is extending the derivation of that earlier work to explicitly cover the non-integer values of the parameter n specifying the polytropic index (calculations in Ref.[5] were only done for integer n and an analytical continuation was conjectured). The extension to the case of a continuous variable n in the present paper is done by using a technique of choosing suitable integration contours in the integral representations of solutions to the equations for the relevant instantons.
While the paper may seem a simple extension of an earlier work, I believe that the topic is sufficiently important and the result is of fundamental theoretical nature, which, in my opinion, justifies its publication in SciPost Physics. It also meets all the general acceptance criteria for publication (clarity, proper citations, etc.) as listed on the journal page. I therefore recommend the paper for publication.
There are only three minor comments that the author may consider to correct (see section "requested changes" below)
Requested changes
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?
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.
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.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)