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Dissipative dynamics in the massive boson limit of the sine-Gordon model
by Ádám Bácsi, Catalin Pascu Moca, Gergely Zaránd, Balázs Dóra
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Submission summary
| Authors (as registered SciPost users): | Ádám Bácsi |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2108.05865v1 (pdf) |
| Date submitted: | 2021-08-20 07:00 |
| Submitted by: | Bácsi, Ádám |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the dissipative dynamics of one-dimensional fermions, described in terms of the sine-Gordon model in its massive boson or semi-classical limit, while keeping track of forward scattering processes. The system is prepared in the gapped ground state, and then coupled to environment through local currents within the Lindblad formalism. The heating dynamics of the system is followed using bosonization. The single particle density matrix exhibits correlations between the left and right moving particles. While the density matrix of right movers and left movers is translationally invariant, the left-right sector is not, corresponding to a translational symmetry breaking charge density wave state. Asymptotically, the single particle density matrix decays exponentially with exponent proportional to $-\gamma t|x|\Delta^2$ where $\gamma$ and $\Delta$ are the dissipative coupling and the gap, respectively. The charge density wave order parameter decays exponentially in time with an interaction independent decay rate. The second R\'enyi entropy grows linearly with time and is essentially insensitive to the presence of the gap.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-10-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2108.05865v1, delivered 2021-10-13, doi: 10.21468/SciPost.Report.3669
Strengths
1- Exact analytical solution of the problem and derivation of results
2- Detailed presentation of the behaviour of physical quantities
Weaknesses
1- Limited discussion of physical significance of the chosen protocol and interpretation of the results
2- Calculations not presented in full detail / replaced by references to earlier publications involving very similar calculations
Report
This work studies the dissipation dynamics of the sine-Gordon in a special limit, the free massive boson limit. Dissipation is accounted for in the form of the Lindblad equation where the jump operators are chosen as the currents of the noninteracting massless model. The initial state is the ground state of the closed system in the same special limit. The dynamical equations can in this case be solved exactly analytically using standard linear transformation techniques presented in earlier works studying technically similar problems ([25,41] by some of the same authors). The main results are expressions for the time evolution of densities, correlation functions and entropies, whose behaviour as functions of space and time is analysed in detail, extracting their scaling laws.
This work is expected to serve as a report of exact results on the increasingly studied subject of dissipation in a many-body model of significant interest. These should be useful for comparison in future numerical studies of the problem in the truly interacting regime of the model or in its many applications in condensed matter and cold-atom physics. The presentation of the paper is clear, especially the discussion of the scaling of physical quantities which is very detailed. Nevertheless in several places the justification of certain choices or steps of the calculation is not included in the text but delegated to the earlier works [25,41]. A weak point is also the absence of an in-depth discussion of the physical significance and interpretation of the results, even though I appreciate that this is partly due to that the Lindblad equation describes dissipation in somewhat abstract terms rather than referring to a concrete dissipation mechanism.
Based on the above and taking into account the acceptance criteria, I recommend publication in SciPost Physics Core instead of SciPost Physics. I would also ask the authors to take into account the following proposed changes.
Requested changes
1- The authors refer to the special limit of the sine-Gordon model they study as the 'massive boson' limit, but this is not a standard or unambiguous term: the sine-Gordon model is an interacting bosonic model and it's massive in a large part of its phase diagram, not only in this special limit. The term 'free' or 'quadratic massive boson' limit is more accurate.
2- As already mentioned, the main thing that is missing from this work is a discussion of the physical significance and interpretation of the results. Why is this choice of jump operators natural? How do the results match with expectations for a system that heats up? Even if some of this discussion is included in earlier works it is worth including and specialising it here. Also, given that the sine-Gordon model is only a low-energy approximation of the systems it is applied on and since the heating process results in higher and higher energy states, it is worth warning that in such systems the sine-Gordon description will eventually break down and the large time dissipative dynamics will be controlled by factors beyond that.
3- Including an appendix with a somewhat more detailed outline of the calculations and intermediate formulas would help to make the manuscript self-contained.
Minor points:
4- Motivation for the present work comes partly from Refs. [26, 27] which are about black hole evaporation / Hawking radiation. What is the connection to the problem of dissipation in the present settings?
5- Another motivation comes from the experimental observations of Ref. [28] and the theoretical study of Ref. [29]. It's worth however noting that some recent theoretical works (arXiv:2010.11214, arXiv:2012.05885) suggest a different explanation of that experiment (parabolic trap).
6- After Eq. (17): For short distances [...] the gap has no effect on the dynamics". There is no time involved in the expression this sentence refers to, it's a ground state equal-time correlator. So I think the authors meant to write something else not 'dynamics' here.
7- There are a few minor grammar errors or typos (e.g. missing articles, non-capital letters in titles in the biliography).
Report #1 by Anonymous (Referee 1) on 2021-9-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2108.05865v1, delivered 2021-09-30, doi: 10.21468/SciPost.Report.3592
Report
The article studies the dissipative dynamics, implemented via a Lindblad-type of coupling to the current, of the one-dimensional Klein-Gordon model. The latter is frequently referred to as the massive boson limit of the sine-Gordon model; while this is technically correct is somewhat leaves the impression that typical features of the sine-Gordon theory are captured which, in my opinion, are not. In particular, the defining soliton excitations of the sine-Gordon model are absent in the considered approximation. It would be nice to comment on the expected effects of keeping the solitons in the dynamics.
Nevertheless, the manuscript addresses a relevant question, which, due to considering the massive boson limit, can be analysed using standard methods. In both the applied techniques and the considered setup the manuscript is a direct generalisation of Reference [25], which treated the same question in the massless limit of the Klein-Gordon model, namely the usual Luttinger theory. In fact, this reference provides the main result for comparison, which is performed in reasonable detail.
The manuscript is written in a clear way, containing sufficient details and references. Thus I conclude that it meets the general acceptance criteria of SciPost. Given that the manuscript is a direct generalisation of Reference [25], I am not sure whether it also satisfies the expectations like "above-the-norm degree of originality". I certainly do not believe that the work satisfies any of the acceptance criteria of SciPost Physics, like a "groundbreaking theoretical/experimental/computational discovery".
Thus while I cannot recommend publication in SciPost Physics, the manuscript may be suitable for a more specialised journal like SciPost Physics Core.
Requested changes
Please address the following points:
1. Is it possible to analyse the problem also using numerical approaches by, for example, considering the mentioned XXZ chain?
2. A discussion about the role of solitons should be added. Is there any expectation about their effect on the dynamics? What would be the technical problems when including solitons?
3. For the dynamics of closed sine-Gordon/Klein-Gordon systems the References [3,4] are given. Are these the only ones, or the most representative ones?