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Magnetisation and Mean Field Theory in the Ising Model

by Dalton A R Sakthivadivel

This is not the current version.

Submission summary

As Contributors: Dalton A R Sakthivadivel
Preprint link: scipost_202108_00058v1
Date submitted: 2021-08-23 18:16
Submitted by: Sakthivadivel, Dalton A R
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
Specialties:
  • Statistical and Soft Matter Physics
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 magnetisation, and the results of the derivation are interpreted graphically, physically, and mathematically. We give an 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.

Current status:
Has been resubmitted



Reports on this Submission

Anonymous Report 2 on 2021-10-11 (Invited Report)

Report

The manuscript discusses well know results of mean field Ising model. The presentation of the manuscript is nice and it highlights important physical concepts associated with the Ising model. I recommend the manuscript for publication, however there are few issues which should need to address before publication.

1) In section I (introduction), 3rd para, where the author describes the general methods of statistical physics (e.g, MFT, RG), there are few lines like:

"In fact, renormalisation group methods are a more reliable way to approximate many systems—in cases where d < 4, or T = T_c.."
"While in four dimensions these finite-size effects are not trivial [18], mean field results generally hold for d ≥ 4, with d = 4 the ‘upper critical dimension.’ "

Since the discussions in this section is completely generic, not model specific, the author should use the term upper critical dimension below which the RG results are relevant rather than explicitly writing d < 4, which is actually the upper critical dimension of Ising model. Author should need to rephrase such sentences in order to make these discussions more general.

2) In a line of the same paragraph author says "...mean field results generally hold for d ≥ 4, with d = 4 the ‘upper critical dimension.’" I would like to mention that at the upper critical dimension, logarithmic modulations of the mean field results are generally expected, which are rather difficult to detect in numerical calculations due to the logarithmic nature of the corrections. However, such logarithmic modulations can be found analytically. Author should address this issue in the text.

3) In subsection II-B (Deriving a mean field model by variational methods), there are few relations e.g., 13, 14, where the author shows product over the index 'i' but in the expressions there are no terms which have index 'i'. Such things can be avoided by introducing Trace over the spin variables.

4) In subsection II-E (Broken symmetry and spin configuration), the author skips a minus sign in the Hamiltonian. Though it will not alter any physics of the Ising system, to make consistent with the earlier subsections, the author may put a minus sign in the Hamiltonian in subsection II-E.

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Author:  Dalton A R Sakthivadivel  on 2021-10-13  [id 1841]

(in reply to Report 2 on 2021-10-11)

The author thanks the second reviewer for their comments. Indeed, the scope of the sentences highlighted in point 1 was unclear, and has been clarified. Additionally, point 2 has been addressed by adding elaborating on the logarithmic corrections needed at $d=d_c$. Regarding point 3 -- I hesitate to introduce trace notation, on the grounds it might be pedagogically unhelpful. However, I fully agree it is useful to clarify the extraneous indices, and so I have included a statement to this effect. Finally, I have addressed the inconsistency in point 4 by negating the relevant term.

Cheers,

Dalton Sakthivadivel

Anonymous Report 1 on 2021-9-15 (Invited Report)

Report

Report on the article “MAGNETISATION AND MEAN FIELD THEORY IN THE ISING MODEL” by Dalton A R Sakthivadivel

Bogoliubov inequality is a variant of what is widely known as Rayleigh-Ritz variational principle in quantum mechanics. This article derives this inequality in Sec. A, applies it to Ising model, and derives the mean field equation for magnetisation (using Jensen’s inequality) in Sec. B, and then provides standard analysis of the mean field equation in Sec. C and Sec. D.

I wonder why any student or researcher should take the trouble of going through this unnecessarily lengthy path to obtain the mean field equation for Ising model. This derivation does not also illustrate the basic physics of mean field approximation. The only merit of this derivation is that it proves that the mean field solution minimises the actual (un-approximated) free energy also.

In my opinion, this article will not perhaps be of much use, and hence may be rejected.

I do NOT recommend publication of this article in SciPost

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Author:  Dalton A R Sakthivadivel  on 2021-09-20  [id 1767]

(in reply to Report 1 on 2021-09-15)

Generally one would not be able to respond to a recommendation of rejection, and I am unsure of what purpose it would serve, but due to the staggered posting of reports I am given the opportunity to.

It is possible there is a disagreement in philosophies. I do not dispute that many of the calculations in the paper are somewhat protracted. To have done otherwise would not have aligned with my approach. In the spirit of what it is to be a set of ‘lecture notes,’ the paper describes in detail not only the performance of these standard calculations, but their motivation. The aim is not towards reference work, but exposition.

"I wonder why any student or researcher should take the trouble of going through this unnecessarily lengthy path to obtain the mean field equation for Ising model." On the contrary, the purpose of a set of lecture notes is to serve as a set of pedagogical tools, aimed at students encountering a subject with perhaps little prior knowledge; the level of detail is commensurate with such a situation. Throughout the paper, any calculation is justified by some other bit of theory before it is employed. This is meant to illuminate how and why mean field theory is applied to the Ising model, not simply show it being done. On the other hand, I would not expect a researcher to spend 19 pages deriving a mean field model before employing it in more sophisticated results.

"This derivation does not also illustrate the basic physics of mean field approximation." Whilst this could be elaborated on, I don't entirely agree with the claim -- in the first section, MFT is described explicitly, and the mean field model is later related to the definition of magnetisation as 'ensemble dynamics' (via the order parameter) in the final section.

Cheers,

Dalton Sakthivadivel

Author:  Dalton A R Sakthivadivel  on 2021-10-13  [id 1842]

(in reply to Report 1 on 2021-09-15)

The author thanks the first reviewer for their comments. To address this feedback I have included further commentary about the motivation for describing collective phenomena, a description of fluctuations in statistical physics more broadly as well as MFT, and some additional remarks about the order parameter. The author hopes this is sufficient to address the first reviewer's concerns about the focus of the manuscript on the physics of MFT.

Cheers,

Dalton Sakthivadivel

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