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

### Submission summary

 As Contributors: Fred Hucht Arxiv Link: https://arxiv.org/abs/1803.10155v3 Date accepted: 2019-08-20 Date submitted: 2019-08-15 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.

Published as SciPost Phys. 7, 026 (2019)

Dear editors,

we thank both referees for their comprehensive reports and changed the manuscript accordingly. Especially report 1 was really helpful and increased the readability of the manuscript. We made all requested changes as recommended, see below, with one exception: The sentence "Only recently..." on page 2, where we commented on the connection between the transfer matrices derived within the dimer method [18,19] and the spinors in [9], was reformulated and extended differently from the referee's recommendations. It now reads:

"Only quite recently, a connection between these two methods was established [18,19], which reduces the Pfaffians emerging in the dimer method to matrices corresponding to the spinor picture even for arbitrary couplings, therefore preserving the possibility to apply arbitrary boundary conditions in both directions [20]. Note that this correspondence goes beyond the simpler case of translational invariant couplings, where both methods are known to lead to the same 2 × 2 matrices."

### List of changes

As requested in Report 1:
o) Changed the word "according" throughout the manuscript.
o) Rewrote the statement "Only recently..." on page 2 to be more precise.
o) Reformulated the sentence page 3, line 4 "Other experiments...".
o) Changed "As the..." to "The..." on page 3, par 2.
o) Added reference to McCoy & Wu [16] on page 4, sec 2. Note that this reference was already given on page 3.
o) Added reference to McCoy & Wu [17] to chapter "Dimers".
o) We fixed all typos: "spacial" -> "spatial" on p.2, "extend" -> "extent" on p.13, etc., see "Requested changes".
o) Changed the phrase "the subsequent part of this paper" to "a subsequent paper" as recommended.

Other changes:
o) Changed $\partial_{\Phi}$ to $\frac{\partial}{\partial\Phi}$ in (4.24).
o) Updated references.

### Submission & Refereeing History

Resubmission 1803.10155v3 on 15 August 2019
Resubmission 1803.10155v2 on 25 June 2019
Submission 1803.10155v1 on 28 March 2018