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Topological Data Analysis of Monopole Current Networks in $U(1)$ Lattice Gauge Theory
by Xavier Crean, Jeffrey Giansiracusa, Biagio Lucini
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Submission summary
| Authors (as registered SciPost users): | Xavier Crean · Jeffrey Giansiracusa · Biagio Lucini |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2403.07739v3 (pdf) |
| Code repository: | https://doi.org/10.5281/zenodo.10806185 |
| Data repository: | https://doi.org/10.5281/zenodo.10806046 |
| Date submitted: | 2024-07-22 12:06 |
| Submitted by: | Crean, Xavier |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
In $4$-dimensional pure compact $U(1)$ lattice gauge theory, we analyse topological aspects of the dynamics of monopoles across the deconfinement phase transition. We do this using tools from Topological Data Analysis (TDA). We demonstrate that observables constructed from the zeroth and first homology groups of monopole current networks may be used to quantitatively and robustly locate the critical inverse coupling $\beta_{c}$ through finite-size scaling. Our method provides a mathematically robust framework for the characterisation of topological invariants related to monopole currents, putting on firmer ground earlier investigations. Moreover, our approach can be generalised to the study of Abelian monopoles in non-Abelian gauge theories.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Please accept the resubmission of our paper 'Topological Data Analysis of Monopole Current Networks in U(1) Lattice Gauge Theory'.
Yours sincerely,
The Authors
List of changes
Changes are in the format (1) brief summary of the referee request, (2) change that the authors have made.
1. More about future non-Abelian application:
We have included a paragraph in the conclusion discussing future applications of our methodology to the non-Abelian case.
2. Change hot/cold nomenclature to low-beta/high-beta:
Throughout the paper, we have changed the nomenclature from hot/cold phase to low-β/high-β phase.
3. Move subsection introducing the Betti numbers to subsection 3.1 and expand section 3:
There are several changes we have made here:
3.1 We have moved the Appendix C that describes background on homology and Betti numbers to a new Subsection 3.1 and created another new subsection 3.2 called `Analysis of Monopole Current Networks` that contains content from the original submission.
3.2 We have created a new Appendix B.3 (a Subsection in the Appendix 3 on Methods) that includes additional details on our computational pipeline. We have gone into much more detail on how we actually compute Betti numbers of the directed graph using giotto-tda. This entails mapping graphs to a 1-dimensional cubical complex with periodic boundary conditions (a data structure that is implemented by the GUDHI library); we have included an explanatory figure.
3.3 In Subsubsection 3.2.1, we have rephrased the description of how we map monopole current networks to directed graphs.
4. Clarify explanation of large network breaking into smaller networks
The picture we describe of a large percolating network in the low beta phase and small distinct networks in the high beta phase is based on the accepted physical picture from the literature. We have added some relevant references to aid the reader. Our comment about the large network breaking into smaller networks as \rho_0 obtains its maximum is a statement that in the critical region the behaviour of \rho_0 corresponds with the accepted physical picture from the literature. We have rephrased the language to clarify the point that at large \beta values sampled configurations are less likely to contain monopole current loops.
5. Reference Kerler et al more explicitly
We have more explicitly mentioned Kerler et al. Ref. [22] in the introduction. We have also included a comment on the fact that the phase transition was found, e.g., in Ref. [29], to exist on a lattice representation of the 4-sphere where non-trivial winding is not possible.
6. Add citation for graph homology independent of coefficient field
We have included an explanation/reference in a footnote.
7. Add variance plots of the observables
We have included the variance plots in the main body of the text with appropriate inset zoom-ins to highlight the peaks in the variance (including error bars) in the critical region.
8. Explain why periodically closed loops come in oppositely oriented pairs
We have included a sentence explaining this.
9. Prefactors in monopole number equation (8)
This was a typo that has now been corrected: Equation (8) now includes the prefactor -1/2.
10. Explain why betti_1 has smaller critical beta than betti_0
We have added a footnote highlighting that whilst E, \rho_0 and \rho_1 are not equal for finite size L, they converge in the physically significant infinite volume limit.
11. Make notation consistent for density of betti_k
Throughout the paper, the notation has been clarified and updated.
12. Make zoom-in plots embedded with diagonal lines
Inset zoom-ins have been added to the plots to highlight the behaviour in the critical region more clearly.
13. Change title to highlight Monopole Current Networks
The title has been changed to “Topological Data Analysis of Monopole Current Networks in U(1) Lattice Gauge Theory”.
14. Sample number N in bootstrap error estimation
We have a made a small edit to the text in Appendix B.2 to highlight the fact that each set S_i, generated by the resampling, consists of N samples.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-8-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2403.07739v3, delivered 2024-08-05, doi: 10.21468/SciPost.Report.9532
Strengths
1. Well written.
2. Clear and sound presentation of the problem and the results.
3. High level of interpretability of the observables defined through topological data analysis.
Weaknesses
I do not see any weaknesses in the current form of the manuscript.
Report
The revised manuscript reads well and addresses the points that I raised during the first round of review. Therefore, I recommend its publication in SciPost Physics.
Below are a couple of details for the authors to consider:
- In the second footnote, I believe that where the authors write $k>2$, it should read $k \ge 2$.
- The label on the y-axis in Fig. 5 should be $\rho_1$ instead of $\rho_0$.
- While the peak signal in Figs. 4 and 6 can be clearly seen, the scattered symbols make these plots a bit difficult to read in detail. Adding lines as a guides to the eye or increasing the size of the markers could be helpful.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2024-8-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2403.07739v3, delivered 2024-08-01, doi: 10.21468/SciPost.Report.9512
Strengths
1. Interesting application of topological data analysis (TDA) to lattice gauge theory.
2. Clear quantitative predictions using homology.
3. State-of-the-art lattice methods.
Report
The authors have provided a major revision of the manuscript, which, from my point of view, forms a substantial improvement compared to its first version. They suitably addressed all my points of critique. In particular, the presentation of the TDA methodology and the discussion of the related results has been significantly improved. The inclusion of the variance curves provides a good addition.
Solely, the manuscript appears now somewhat technical at some points to me, in particular regarding the concepts from algebraic topology. A sentence on the relation between chain complexes and graphs, based on lattice gauge theory configurations, might help guiding the physical intuition in Sec. 3.1.2. Furthermore, in Sec. 3.1.2 I appreciate the argument regarding coefficient field independence of the Betti numbers, but maybe the sentence including the universal coefficient theorem can be included in footnote 2 as well? I am doubtful, whether the anticipated readership of SciPost Physics is aware of the universal coefficient theorem; and the statement of coefficient field independence would then still be included in the main text. Finally, Appendix B.3 including the related new Fig. 8 adds value to the manuscript, but I am confident that the concept of a stratification can be omitted, which is certainly unclear to many physicists.
Yet, I recommend publication of the manuscript in SciPost Physics.
Requested changes
The minor change recommendations, which are to be viewed as suggestions only, have been highlighted in the report.
Recommendation
Publish (meets expectations and criteria for this Journal)