Dissipative dynamics in the free massive boson limit of the sine-Gordon model

Submission summary

 As Contributors: Ádám Bácsi Arxiv Link: https://arxiv.org/abs/2108.05865v2 (pdf) Date submitted: 2021-11-24 11:07 Submitted by: Bácsi, Ádám Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

Abstract

We study the dissipative dynamics of one-dimensional fermions, described in terms of the sine-Gordon model in its free 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.

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--- RESPONSE TO REFEREE REPORT 1 ---

We thank the Referee for the detailed report. Let us start by noting that to the best of our knowledge, it is not clear whether the dissipative sine-Gordon model is exactly solvable or not, unlike its non-dissipative counterpart, which is Bethe Ansatz solvable. Therefore, in this regard, any solution, which is reliable for a certain range of parameters, should be of high value. This is especially true when correlation functions can also be calculated. In this regard, our work represents the first step in this direction to understand the dynamics of the dissipative sine-Gordon model.

As to the comments of the Referee:

1. We agree with the Referee that in principle it would be possible to study the dynamics, we predict, also numerically on the XXZ chain. However, given the nature of the various crossovers and associated lengthscales in the correlation functions, it would be very demanding to catch and identify all these features. We have performed exact diagonalization studies on the XXZ chain up to N=14 sites and found partial agreement with our analytical results. However, due to the small system sizes, these data were inconclusive, therefore we refrain from displaying it in the paper.

2. As to the effect of the solitons, as long as the Luttinger liquid parameter is small, i.e. K<<1, expanding the cosine potential around one of its minima is a rather good approximation and one does not need to worry about tunneling to adjacent minima, as also mentioned in the newly cited papers by Foini et al. and Ruggiero et al. In this limit, the soliton mass is pushed up to very high energies with 1/\sqrt(K) and the solitons practically decouple from the dynamics. However, their effect can most probably be taken into account using a form factor expansion or by using a perturbative expansion in $K$, which is only expected to affect the short time transient dynamics.

3. The references we cite in connection to the sine-Gordon model are certainly not the most ideal papers, so we added some additional references.

--- RESPONSE TO REFEREE REPORT 2 ---

We thank the Referee for his/her detailed report. Before we respond to each requested change, let us emphasize that to the best of our knowledge, it is not clear whether the dissipative sine-Gordon model is exactly solvable or not, unlike its non-dissipative counterpart, which is Bethe Ansatz solvable. Therefore, in this regard, any solution, which is reliable for a certain range of parameters, should be of high value. This is especially true when correlation functions can also be calculated. In this regard, our work represents the first step in this direction to understand the dynamics of the dissipative sine-Gordon model.

1. We thank for raising our attention to this unambiguousness. We changed the terminology accordingly.

2. Using the local current as jump operator is rather natural since it can easily arise from fluctuating gauge fields or vector potentials, which couple naturally to the current. Another natural choice for the jump operator would be the local density, and our calculations can be applied for that case as well with minor modifications, e.g. by replacing K with 1/K. We also explained that strong spatio-temporal suppression of correlation function is in accord with what is expected from a very high temperature thermal state where correlations are suppressed beyond the time and length scale associated to temperature. In connection to the entropies, we mention that their monotonic temporal increase agrees with the physical picture that the system heats up to infinite temperatures upon coupling it to a bath through a hermitian jump operator. However, the late time $\ln(t)$ entropy growth is connected to the infinite dimensional local Hilbert space of the $b_q$ bosons, and is absent in lattice models with a finite dimensional local Hilbert space (i.e. for fermions or hard core bosons). In this limit, the sine-Gordon description will eventually break down and the entropies are expected to saturate to their maximal values, determined by the finite size of the Hilbert space.

3. A new appendix is added, where the main steps for calculating the correlations functions and the entropies are highlighted.

4. The Hawking radiation is presumably not the most ideal motivation for this study, we replaced it by "targeted cooling into topological states from arbitrary initial states".

5. We added these two citations.

6. We thank for the remark and corrected the expression accordingly.

7. We tried to correct the reference list. Some issues must be related to some bibtex setting, everything looks perfect in our .bib file. This will certainly be fixed in the published version.

List of changes

- The terminology of the limit is clarified as "free massive boson limit of the sine-Gordon model" throughout the whole manuscript including the title.
- In the second paragraph of the introduction, targeted cooling is added as related phenomenon, new references are also cited.
- Between Eqs. (5) and (6), we added a new paragraph about expected effect of solitons.
- Before Eq. (7), explanation is added about the choice of jump operators.
- Eq. (9a) is reformulated.
- New appendices are added to explain the derivation of Eqs. (13), (24) and (33).
- The sentence after Eq. (17) is corrected.
- Additional interpretation of the results is added to the end of the second last paragraph of Sec. III.
- Remark about numerical simulation is added to the last paragraph of Sec. III.
- Additional paragraph to the end of Sec. IV.

Submission & Refereeing History

Resubmission 2108.05865v2 on 24 November 2021
Submission 2108.05865v1 on 20 August 2021

Reports on this Submission

Report

The authors have taken into account the remarks by the referees. Thus the manuscript can be published in its current form. However, given the acceptance criteria of SciPost Physics, my recommendation that the manuscript is suitable for a more specialised like SciPost Physics Core remains.

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