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Magnetisation and Mean Field Theory in the Ising Model
by Dalton A R Sakthivadivel
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Dalton A R Sakthivadivel |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2102.00960v4 (pdf) |
| Date accepted: | 2021-11-09 |
| Date submitted: | 2021-10-14 04:37 |
| Submitted by: | Sakthivadivel, Dalton A R |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.
Author comments upon resubmission
List of changes
In this resubmission, the following revisions have been made:
Expanded commentary about
- Collective phenomena (Introduction)
- Fluctuations in MFT and statistical physics more broadly (Introduction)
- Order parameters (IIE)
Amendments to
- Discussion of renormalisation group and critical dimension (Introduction)
- Notation in some areas
All pursuant to reviewer comments. Additionally, a link to an interactive surface plot hosted by the author on GeoGebra has been included, for pedagogical value.
Published as SciPost Phys. Lect. Notes 35 (2022)
Reports on this Submission
Report #3 by Anonymous (Referee 1) on 2021-10-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2102.00960v4, delivered 2021-10-27, doi: 10.21468/SciPost.Report.3748
Report
This is a report on ``Magnetisation and Mean Field Theory in the Ising Model''
by Dalton A R Sakthivadivel. In this paper, the author presented a pedagogical tutorial for the mean-field theory of the two-dimensional Ising model. In addition to discussing the standard mean-field theory, the author makes connections with the Bogoliubov inequality. The results and their implications are discussed both physically and mathematically.
This paper has already been reviewed by two referees twice (the second time being a revised version submitted by the author). One referee had asked for some revisions of the originally submitted version, and eventually recommended acceptance and publication of the revised manuscript, based on the author's response and the revisions. The other referee remained unconvinced about the utility of this work and persisted with his/her recommendation of rejection.
I have carefully read the versions of the paper as well as the referee comments and the corresponding author responses. While it is true that the manuscript largely has known, standard results, I believe it presents the results in a way that might be of some use to students/younger researchers getting initiated into this field.Therefore, on the balance, I recommend publication of the revised manuscript without any change.
Report #2 by Anonymous (Referee 2) on 2021-10-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2102.00960v4, delivered 2021-10-16, doi: 10.21468/SciPost.Report.3687
Report
I read the article again, as much objectively as possible. I am sorry to say that I found no reason to alter my decision of rejection.
Mean field theory starts with the basic assumption that the magnetic field acting at any site is the average magnetic moment. With this assumption, one can derive a transcendental equation for magnetic moment (Eq. (14) in the article) following a simple argument, as can be found in any standard text book. The author however arrives at this equation by looking for the value of magnetic moment which minimises the variational free energy. This alternative approach (i) uses “Bogoliubov inequality”, in addition to standard treatments; (ii) is lengthy (iii) adds to our understanding ONLY the knowledge that the conventional mean-field value of magnetic moment minimises the variational free energy. Although I understand that the philosophy of different scientists may be different, I strongly feel that there is no reason why the point (iii) could be helpful to any beginner (student) or researcher (in other areas of physics).
Hence, I stick to my decision of rejection.