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Generalized hydrodynamics of classical integrable field theory: the sinh-Gordon model
by Alvise Bastianello, Benjamin Doyon, Gerard Watts, Takato Yoshimura
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Submission summary
| Authors (as registered SciPost users): | Alvise Bastianello · Benjamin Doyon · Gerard Watts |
| Submission information | |
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| Preprint Link: | http://arxiv.org/abs/1712.05687v2 (pdf) |
| Date submitted: | 2018-02-01 01:00 |
| Submitted by: | Bastianello, Alvise |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Using generalized hydrodynamics (GHD), we develop the Euler hydrodynamics of classical integrable field theory. Classical field GHD is based on a known formalism for Gibbs ensembles of classical fields, that resembles the thermodynamic Bethe ansatz of quantum models, which we extend to generalized Gibbs ensembles (GGEs). In general, GHD must take into account both solitonic and radiative modes of classical fields. We observe that the quasi-particle formulation of GHD remains valid for radiative modes, even though these do not display particle-like properties in their precise dynamics. We point out that because of a UV catastrophe similar to that of black body radiation, radiative modes suffer from divergences that restrict the set of finite-average observables; this set is larger for GGEs with higher conserved charges. We concentrate on the sinh-Gordon model, which only has radiative modes, and study transport in the domain-wall initial problem as well as Euler-scale correlations in GGEs. We confirm a variety of exact GHD predictions, including those coming from hydrodynamic projection theory, by comparing with Metropolis numerical evaluations.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2018-3-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1712.05687v2, delivered 2018-03-09, doi: 10.21468/SciPost.Report.369
Strengths
1. The subject of the paper is currently in the focus of research.
2. New analytic results are derived for the classical sinh-Gordon model.
3. The paper provides direct evidence for a number of conjectured expressions and methods (see Report).
Weaknesses
1. The clarity of the presentation could be slightly improved.
Report
The so-called Generalised Hydrodynamics (GHD) is a young theory that describes large (hydrodynamic, "Eulerian") scale behaviour of integrable systems, originally developed in the quantum case. The paper contributes to this fresh field of research by generalising GHD to classical field theories, which offers a way of testing some of its ideas and explicit expressions. Comparing the predictions with independent numerical simulations is easier in the classical case (especially in the continuum). The comparison provides evidence that the GHD formalism is valid even for classical field theories having radiative modes, which is an interesting new result.
In particular, the numerical tests verify in the classical domain
- the GHD formula for correlation functions,
- the so-called hydrodynamic projection method that allows for the computation of correlations of operators that are not conserved densities or associated currents,
- a recently proposed expression for GGE expectation values of vertex operators.
Based on this, I recommend publication of the manuscript.
I have the following suggestions and questions to the authors which in my opinion could further improve the paper:
1. Perhaps the physical situations considered in Sec. 3.2 and Sec. 3.3 could be stated more clearly and explicitly, even in the Figure captions.
2. The authors may want to consider extending the content of Sec. 3.3 to the case of a GGE state or even to the partitioning protocol.
3. Can the V-functions of vertex operators be derived from the semiclassical limit of form sinh-Gordon form factors?
4. The method for the initial state preparation is not given, only a reference is cited. I think it deserves a couple of sentences at least in the Appendix.
5. It is not immediately clear why an integrable discretisation of the field equations leads to smaller statistical errors.
Some typos:
- in the 3rd paragraph of the Introduction the parenthesis of "(see also" is not closed.
- it seems that a factor of g is missing from the argument of cosh in Eqs. (48) and (49).
- the punctuation of the sentence around Eqs. (52) and (53) is not clear.
Requested changes
1. Clearer motivation and presentation of the setups studied in Sec. 3.2 and 3.3.
2. Some details on the initial state preparation.
Report #1 by Anonymous (Referee 1) on 2018-3-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1712.05687v2, delivered 2018-03-05, doi: 10.21468/SciPost.Report.364
Strengths
1. The subject considered is timely and interesting.
2. The problem is original and relevant to the development of further research.
3. The results are solid and clearly stated.
Weaknesses
1. I find that some notions are not sufficiently defined/explained.
2. In some aspects, I think the clarity of presentation should be improved.
See report for details.
Report
The paper applies the newly introduced formalism of Generalised Hydrodynamics (GHD) to classical field theory, more specifically the sinh-Gordon model. The GHD itself is partly based on conjectures, so besides extending the formalism to a new class of models, this work also provides an important test of its validity. In addition, the results also provide a test of the validity of a recursive formula for the one-point function of exponential operators. These results warrant the publication of this paper as Tier II, after the authors considered the suggestions for improvement listed below.
Requested changes
1. The authors use the notion "Euler scale", but it's not defined anywhere. Presumably this is meant to ebe the scale on which macroscopic hydrodynamics is valid, but this is unclear. Also, is it possible to give some more specific characterisation of this regime (e.g. some minimum length scale specified in terms of the microscopic dynamics)?
2. What do the authors mean when writing "Euler-scale correlation functions are more precisely obtained by averaging over fluid cells"? Is this a more detailed and precise definition of the correlation functions, or it is the case that numerical precision is enhanced by the averaging?
3. In the paragraph after eqn. (36) the authors specify that the starting configurations involve only chemical potentials coupled to the charge Q1, or to Q1 and Q3. This is however not sufficiently emphasized and the use of continuous text with inline formulae make the passage less readable. I would also like to ask whether in the GGE case inclusion of unequal beta1 on the two sides, and/or a further charge (say Q5) would present much difficulty. I think that including such cases would make the results substantially more convincing and general.
4. With respect to Figure 3, could the authors provide a quantitative measure for the deviation from the free result to demonstrate that the interacting case provides a better fit (a possibility would be the integrated deviation ratio they use to quantify the matching between the numerical simulation and the interacting GHD). Alternatively, is there a choice for the parameters beta and g such that the difference is more observable then for the case they plot in Figure 3?
5. In the first paragraph of Section 3.3, I find that the use of the inline formulae make the text much less readable (maybe this is subjective, though).