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Rapid detection of phase transitions from Monte Carlo samples before equilibrium
by Jiewei Ding, Ho-Kin Tang, Wing Chi Yu
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Submission summary
| Authors (as registered SciPost users): | Wing Chi Yu |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202203_00027v2 (pdf) |
| Code repository: | https://github.com/ParcoDing/Rapid-detection |
| Data repository: | https://github.com/ParcoDing/Rapid-detection |
| Date submitted: | 2022-06-27 10:23 |
| Submitted by: | Yu, Wing Chi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
We found that Bidirectional LSTM and Transformer can classify different phases of condensed matter models and determine the phase transition points by learning features in the Monte Carlo raw data before equilibrium. Our method can significantly reduce the time and computational resources required for probing phase transitions as compared to the conventional Monte Carlo simulation. We also provide evidence that the method is robust and the performance of the deep learning model is insensitive to the type of input data (we tested spin configurations of classical models and green functions of a quantum model), and it also performs well in detecting Kosterlitz–Thouless phase transitions.
List of changes
1. The detailed description of our method in Sec. 1 is removed and the details are left to Sec. 2 to avoid redundancy.
2. The discussion of the embedding layer in the second paragraph of Sec. 2 is modified.
3. The meaning of one (quantum) MC step is now defined in the last paragraph of Sec. 2.
4. Solid lines in Figure 2 and Fig. 4 are replaced by the tanh fits of the corresponding machine output.
5. Figure 3 is replaced by transition temperatures extracted from tanh fit of the machines’ output, and the corresponding discussion on P.8 is also modified accordingly. The original Fig. 3 where the transition temperatures were extracted from a linear fit is moved to Appendix B.
6. Further discussions on the large fluctuations observed in the vicinity of the critical point in the Hubbard model (Fig. 4(d3)) is added in the second last paragraph of p.12 and Appendix C.
7. Table 1 is added to compare the time cost by using conventional methods and our method in detecting the phase transition points and the corresponding discussion is added to the last paragraph of Sec. 4.
8. Results of shuffling the input series of XY model and Hubbard are added to Appendix A.1.
9. Appendix A.2 and Appendix A.4 is added to discuss the machine performance on varying the number of randomly selected sites and the size effect, respectively.
10. Appendix D is added to include the details of the sample collection for obtaining the data in Table 1.
11. The raw data and codes have been made publicly available at the link https://github.com/ParcoDing/Rapid-detection.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-7-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202203_00027v2, delivered 2022-07-22, doi: 10.21468/SciPost.Report.5441
Report
I thank the authors for the careful discussion and investigation of my questionings. I am happy with the responses and consider the paper to be in good shape. I recommend it for publication.
Below I briefly comment each of the authors replies to the issues I had raised.
#1: Thank you for including Table 1 and appendix D. These add valuable information to the manuscript.
#2: Thank you for clarifying this.
#3: In appendix A.4 the authors wrote that "the extracted Tc is insensitive to the system size and agree well with the theoretically predicted value". I have trouble understanding this result, since I would expect the Tc estimate to be size-dependent. But perhaps such dependence is hidden by the error of the method's Tc estimate. Anyway, I am happy with the discussion of this issue added by the authors. Further investigations of this matter may be addressed in subsequent works.
#4: Thank you for this change.
#5: Perfect.
#6: Thank you for improving this figure. Looking at its new version, it looks like the Bi-LSTM estimate is largely unchanged for 20 or more MC steps with very small error bars. I wonder if this means the Bi-LSTM estimate can be systematically improved with the number of MC steps, even if in this case the converged estimate is slightly biased. Again, a smaller question that might be worth further investigation in future works.
#7: This is valuable information. Thank you for addressing it.
#8: Thank you.
#9: Thank you for further investigating this matter and more carefully discussing it in the text. I am perfectly happy with it. Again, it would be interesting if future works attempt to find ways of circumventing this issue and improve critical point estimates.
#10: Thank you for adding this information and discussion.
#11: Thank you.
#12: Perfect! Thank you.
#13: The text reads much better now. I still came across a couple of minor spelling mistakes that I list bellow.
Requested changes
As I have just mentioned, a few spelling issues remain:
1 - There are a couple of occurrences of the expression "for examples" across the text.
2 - In introduction, P-2, paragraph 3, where the text reads "Our method being discuss here..." should be read "discussed".
3 - At the last paragraph of the introduction, P-3, the sentence "The paper is organised as the followings." should read "The paper is organised as follows.".
4 - First paragraph of P-8 reads "The temperature corresponds to a fitted value...". My interpretation is that it should read "The temperature that corresponds to a fitted value...", otherwise it might hinder the readers understanding.
5 - First paragraph of P-11 reads "the transnational symmetries" while it should read "the translational symmetries".
6 - Second paragraph of P-11 reads "as compared to the method of calculating the order parameters and supervised learning". To me this sentence sounds a bit confusing so I would suggest it be rewritten into something like "as compared to the traditional method of estimating the transition from order parameters and to the supervised learning method".
7 - First paragraph of Conclusion, P-13, reads "before equilibrium and located the critical points" but should read "before equilibrium and locate the critical points".
8 - In Appendix A.4 the text reads "the step size used here is m = 10". This refers to the number of steps taken from the beginning of the MC walk. The way in which it is written might cause confusion on distracted readers (with for example, "time step" of projector quantum MC methods). It could thus perhaps be improved.
Report #1 by Anonymous (Referee 2) on 2022-7-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202203_00027v2, delivered 2022-07-17, doi: 10.21468/SciPost.Report.5410
Report
1- The paper now is well structured and redundancies are removed.
2- The authors address the problem with the linear regression. The extraction of the transition points is now much clearer. The guide to the eye is now replaced with the actual fit. This strengthen the data and gives an idea about the precision of the fit by only looking at the plots.
3- Thank you very much for clarifying this.
4- The explanation and the structure revision of the article now makes everything clear.
5- This is a good comparison and increases the significance of the work.
6- See requested changes. The rest of the article reads much better now.
7- Thank you for sharing this.
I recommend publication and learned a lot from the paper.
Requested changes
1- Last sentence of the second last paragraph on page 3 needs to be reordered or rewritten. No big deal.