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Exact solution for the Lindbladian dynamics for the open XX spin chain with boundary dissipation
by Kohei Yamanaka and Tomohiro Sasamoto
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Submission summary
| Authors (as registered SciPost users): | Kohei Yamanaka |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202105_00008v2 (pdf) |
| Date submitted: | 2022-12-22 09:18 |
| Submitted by: | Yamanaka, Kohei |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We obtain exact formulas for the time-dependence of a few physical observables for the open XX spin chain with Lindbladian dynamics. Our analysis is based on the fact that the Lindblad equation for an arbitrary open quadratic system of N fermions is explicitly solved in terms of diagonalization of a 4N×4N matrix called structure matrix by following the scheme of the third quantization. We mainly focus on the time-dependence of magnetization and spin current. As a short time behavior at a given site, we observe the plateau regime except near the center of chain. Basic features of this are explained by the light-cone structure created by propagations of boundary effects from the initial time, but we can explain their more detailed properties analytically using our exact formulas. On the other hand, after the plateau regime, the magnetization and spin current exhibit a slow decay to the steady state values described by the Liouvillian gap. We analytically establish its 1/N^3 scaling and also determine its coefficient.
Author comments upon resubmission
We have revised the manuscript "Exact solution for the Lindbladian dynamics for the open XX spin chain with boundary dissipation" according to the referees' comments. We believe we have been able to answer to almost all the comments and to produce an improved version of the paper. We apologize that it has taken so long to revise the paper.
List of changes
• We clarified technical novelty compared to previous studies in section 2.
• We completely rewrote section 4. Two main changes are the followings.
• We added detailed discussion of the plateau regime by performing an asymptotic analysis of integral representations of physical quantities.
• We examined the large N behavior of the Liouvillian gap more in detail analytically and were able to determine the coefficient in front of the 1/N^3 scaling.
• We correctted minor typos/modify sentences.
Current status:
Reports on this Submission
Strengths
1 - technical novelty
2 - derivation of novel exact results
Weaknesses
1 - the phenomena described in this paper are unsurprising and can be expected based on very simple physical arguments
Report
After revision, the authors have extensively modified their manuscript, in particular starting from page 14. They have removed some figures (without loss of clarity) and also made the physical interpretation of their results more clear.
They have also clarified their statements concerning the degree of novelty of their technical results.
Requested changes
Minor comments/typos:
Typo Eq. 4? \varepsilon_R -> \varepsilon_L in L_2 ?
Same in Eq. 8 ?
Typo page 13 : Lindbradian -> Lindbladian
Report #2 by Anonymous (Referee 3) on 2023-2-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202105_00008v2, delivered 2023-02-11, doi: 10.21468/SciPost.Report.6720
Strengths
Sound calculations compared with numerics
Weaknesses
Written in a rather technical way
Most findings are expected
Report
In the revised version the authors (i) develop a method for analytically determining the post-quench dynamics of the magnetisation and spin current of the boundary-driven XX chain. Subsequently, the authors use their method to obtain two concrete results (ii) the height of the magnetisation plateau observed in intermediate-time dynamics, and (iii) the coefficient of the long-time relaxation rate to the steady-state. This revised version of the manuscript is significantly different from the previous one, having in common only the result (i).
Regarding novelty
I maintain my main view. Despite not being particularly surprising or original, the phenomenology presented in the paper requires serious technical effort.
Results (i) are relatively straightforward, but technically involved, and are useful to other researchers looking at analytical aspects of post-quensch dynamics in open systems.
Results (ii) and (iii) are novel. They also seem to be sound and were tested against numerical simulations.
Regarding the clarity and organization of the paper.
I maintain my previous viewpoint. There was no substantial change in this aspect.
Main criticism.
Most of my previous criticisms no longer apply since the paper was substantially modified.
I believe that the authors now explore the power of their analytical solution to obtain results (ii) and (iii). Therefore, my previous criticism was addressed.
The main critiques I have now are related to the presentation since I still find the paper hard to read. For example, it would be helpful to have the final expression of $\Delta$ in Eq.(61) terms of physical quantities $\varepsilon_{R/L}$, $J$, etc. However, to understand their formula, the reader must carefully examine the calculations to find the definitions of $l$ and $r$ below Eq (51). Putting the final result in terms of the physical quantities defined at the beginning of the manuscript could highlight physical aspects and help a non-technically inclined reader to still be able to find some useful information. As another example, b defined below Eq (51) is never used thereafter. This gives the impression that the authors passed the calculations from their notebook to the paper without thinking about the most effective way of presenting them to the reader.
Other points:
I believe there is a problem with the caption of Fig 4. Eq (59) should refer to Eq (58)
Assessment.
I recommend the manuscript for publication. However, I strongly advise the authors to improve the readability of the manuscript.
Author: Kohei Yamanaka on 2023-02-17 [id 3363]
(in reply to Report 2 on 2023-02-11)
We thank the referee for his/her careful reading of our manuscript and for giving useful comments. We have carefully checked all the comments by the referee on our manuscript and improved it. A version of our new draft, in which all changes are displayed in red, is also attached.
The referee wrote:
<Weakness>
-Written in a rather technical way
<Regarding the clarity and organization of the paper>
-I maintain my previous viewpoint. There was no substantial change in this aspect.
<Main criticism>
-The main critiques I have now are related to the presentation since I still find the paper hard to read.
Our response:
We have improved several points in our manuscript in order to make our paper more accessible to non-technically inclined readers.
First, the referee wrote:
-For example, it would be helpful to have the final expression of Delta in Eq.(61) terms of physical quantities…
Our response:
In the revised version, we follow the advice and write the final expression of $\Delta$ (61) in terms of the physical quantities $J$, $\varepsilon_L$, and $\varepsilon_R$. At the same time we also made the use of parameters l and r in a more systematic way. Namely, we now define the parameters l and r below Eq.(24) on page 7 where they appear for the first time and then consistently use them except for the final results. We believe that these modifications have simplified several formulas and allowed the final results to be understood by non-technically inclined readers.
Second, the referee wrote:
-As another example, b defined below Eq (51) is never used thereafter.
Our response:
We have rewritten b to j/(4J).
In addition to the above two points, which were pointed out by the referee, we have added a few more explanations about motivations and summaries of our discussions in sections 2 and 3 on pages 4,6, 9, and 11. By this revision, we believe that non-technical inclined readers may understand our aims and results better without reading detailed calculations.
Finally we have removed the following unnecessary abbreviations.
page 2 NEGF and QME
page 8 NE and SE
We hope that this small change also improves slightly readability of our paper.
Other points:
The referee wrote:
-I believe there is a problem with the caption of Fig 4. Eq (59) should refer to Eq (58).
Our response:
We thank the referee for pointing this out. We have corrected this typo as (59)->(58).
Additional changes in this revision.
In Fig. 1, we have rewritten the legends for each line from epsl and epsr to the Greek letters $\varepsilon_L$ and $\varepsilon_R$.
In Eq.(23) we set the numerator of the first factor $q_1^{(k)}$ to unity because we can choose $q_1^{(k)}$ to be an arbitrarily non-zero real number and it would be easier to understand the final result.
Author: Kohei Yamanaka on 2023-02-17 [id 3364]
(in reply to Report 1 on 2023-02-13)We thank the referee for his/her careful reading of our manuscript and for giving useful comments. The only requested changes were a few minor comments/typos. All noticed typos and comments have been fixed.