# Anisotropic scaling of the two-dimensional Ising model I: the torus

### Submission summary

 As Contributors: Fred Hucht Arxiv Link: https://arxiv.org/abs/1803.10155v2 (pdf) Date submitted: 2019-06-25 02:00 Submitted by: Hucht, Fred Submitted to: SciPost Physics Discipline: Physics Subject area: Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and - if present - the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

Dear editors,

we thank both referees for their reports and changed the manuscript accordingly. Furthermore, we submitted the second part of the work [arXiv:1805.00369v2] to SciPost, as suggested by the first referee JHH Perk. We rewrote large parts of the manuscript, updated several figures, and changed the notation at many places on order to increase the readability and simplify the calculation.

### List of changes

o) Equations are now numbered by section
o) Added text and references to Introduction
o) Added several references to the other sections
o) Added a minus sign to H (2.8), and updated the following equations accordingly, such that antiperiodic boundary conditions in the Ising model get the subscript "-".
o) Changed $J_{\beta}^{-}$ to $-J_{\beta}^{-}$ in (2.23)
o) Explicitly defined the dual (2.24)
o) Renamed matrices $\Delta,\Sigma$ to $K^{\pm}$
o) Removed Fig. 4
o) Moved (46a) to (3.2)
o) Simplified (3.25)
o) Removed the incorrect paragraph after (3.25)
o) Simplified (4.35)
o) Changed subsection 4.3 to be section 5

### Submission & Refereeing History

Resubmission 1803.10155v3 on 15 August 2019

Resubmission 1803.10155v2 on 25 June 2019
Submission 1803.10155v1 on 28 March 2018

## Reports on this Submission

### Anonymous Report 2 on 2019-7-12 Invited Report

• Cite as: Anonymous, Report on arXiv:1803.10155v2, delivered 2019-07-12, doi: 10.21468/SciPost.Report.1063

### Strengths

1- The authors obtain new results on the anisotropic Ising model in a toroidal geometry.
2- The simplifications with respect to previous methods should allow the authors to treat more general cases in the future.

### Weaknesses

1- Prior to acceptance, the authors would need to correct a few minor details, mainly having to do with English usage and spelling.

### Report

The authors have responded very thoroughly to the comments on an earlier version. The paper can now be accepted, after the correction of a few minor details.

### Requested changes

1- My list of changes turns out to be a (proper) subset of those requested by the first referee, so the authors can refer to his or her list.

• validity: top
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: good

### Anonymous Report 1 on 2019-7-12 Invited Report

• Cite as: Anonymous, Report on arXiv:1803.10155v2, delivered 2019-07-12, doi: 10.21468/SciPost.Report.1062

### Strengths

Paper gives new results on finite-size scaling limits of the anisotropic Ising model on a torus.

### Weaknesses

See requested changes.

### Report

The resubmitted manuscript has adequate citations and new results added and, as far as I can see, these results are correct. I have gone through a lot of the calculations and even though I would say some things differently, the authors should have the freedom to say those things their own way. I recommend publication after the authors clean up several minor issues, including both confusing and trivial spelling errors. It should not take them much time. This paper can now stand on its own after new results were added on the finite-size scaling functions for anisotropic Ising on a torus.

### Requested changes

First, the word "according" appears many times in the manuscript, where other words like "corresponding", "associated" or "related" fit better in the context and make the meaning clearer.

Page 2, line 16 about: I am not comfortable with the "Only recently ..." statement. The "direct connection" does not occur early in the two papers. Also, there are earlier instances where dimer and spinor calculations for Ising meet, especially for correlation functions. Rolling up the model row-by-row and applying Fourier transform, 2x2 matrices already showed up very early in dimer model free-energy calculations also. A sentence like "It is very interesting that [9] using spinors and [18,19] using dimers end up with the same 2x2 matrices ..." 0r similar seems more appropriate.

Page 2, 3rd line paragraph 3: "spatial" is the preferred spelling; "spacial" is improper usage that has become accepted.

Page 3, line 4: "are" needs to be deleted.

Page 3, paragraph 2: The opening "As" may have to be omitted and the comma before "furthermore" may then be replaced by ";".

Page 4, section 2, line 3: It is fair to say here that McCoy and Wu pioneered this in Chapter XIV of their book based on Phys. Rev. 176, 631 (1968). The generalization here follows their method with the horizontal couplings also varying from row to row.

Pages 5-8: The writing is not clear to people not familiar with the dimer method. They would be required to consult the McCoy-Wu book or similar. It would help these readers, if they were told to consult chapter IV and section 2 of Chapter V of that book for a rather extensive treatment of the method.

Page 13, lines 4 and 5: extend (verb) should become extent (noun).

Page 15, line above (3.17): "scaling functions" should be plural (or "consists" singular).

Page 16: Minor remark (optional): (4.3) is (anti)cyclic, so the determinant can also be found by diagonalizing with Fourier similarity transform.

Page 17, (4.9): At this point it may also be good to remind people of (2.3) for $Z_0^{(p,p)}$. They would have to really seek for it.

Page 20, line 1: "throughout" is one word.

Page 23, bottom line: Replace "Eventually" by "Finally".

Page 24, line below (4.37): Replace "for most" by "foremost".

Page 25, bottom line: Remove comma in "see, Fig. 8".

Page 27, 2 lines below (5.6): Replacing $\partial_x$ by $d/dx$ is clearer.

Page 27, 2 lines from bottom: "Therefor" should be "Therefore,".

Page 28, middle: "the subsequent part of this paper [28]" sound strange. Could be changed to "a subsequent paper [28]", "a companion paper [28]", "a later paper [28]", or similar.

• validity: high
• significance: high
• originality: high
• clarity: good
• formatting: excellent
• grammar: good