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Magnetisation and Mean Field Theory in the Ising Model
by Dalton A R Sakthivadivel
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|As Contributors:||Dalton A R Sakthivadivel|
|Date submitted:||2021-08-23 18:16|
|Submitted by:||Sakthivadivel, Dalton A R|
|Submitted to:||SciPost Physics Lecture Notes|
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.
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Anonymous Report 2 on 2021-10-11 (Invited 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 , 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.
Anonymous Report 1 on 2021-9-15 (Invited 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