## SciPost Submission Page

# Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields

### by Hendrik Hobrecht, Alfred Hucht

#### - Published as SciPost Phys. 8, 032 (2020)

### Submission summary

As Contributors: | Fred Hucht |

Arxiv Link: | https://arxiv.org/abs/1805.00369v3 |

Date accepted: | 2020-01-29 |

Date submitted: | 2019-12-27 |

Submitted by: | Hucht, Fred |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Statistical and Soft Matter Physics |

Approach: | Theoretical |

### Abstract

Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.

Published as SciPost Phys. 8, 032 (2020)

### Author comments upon resubmission

we thank both referees for their comprehensive reports and changed the manuscript accordingly. Both reports were really helpful and increased the readability of the manuscript. We made all requested changes as recommended, see below, with only a two exceptions: Schur reductions are common in mathematical physics, hence we did not cite specific papers, and the term "keyhole integral" is also common in complex integration where cuts are involved, so we did not change this.

Furthermore, we rewrote several sentences in order to increase the readability of the manuscript. A list of changes is given below.

### List of changes

Changes as requested in Report 1:

o) changed the incorrectly used wordings as "according", "converge against", and so on, and fixed many typos

o) added a remark to Chebyshev polynomials near (2.9)

o) better explained the use of subscript "l" and "r"

o) better explained α and β

o) fixed Refs. [47,48]

o) fixed the formulation above (2.25)

o) fixed the contour lines in the complex contour plots Figs. 2, 5, 11, 15. We are grateful to referee 1 for pointing out the ambiguous use of contour lines. Now "the lines of constant absolute value c are shown as black dotted (c < 1), dashed (c = 1) or solid (c > 1) lines, where c are consecutive integer powers of two."

o) replaced ∂x by d/dx throughout the manuscript

o) fixed the wording concerning the sum over m on page 29

o) renamed the staggered boundary condition from "Brascamp-Kunz" to "staggered" throughout the paper. We agree with both referees that "staggered" and "doubly staggered" better reflects these boundary conditions.

Changes as requested in Report 2 (if not already noted above):

o) fixed the spelling error "transition circle" to "transition cycle". Thank you for this hint.

o) added references regarding the equivalence of Dirichlet, staggered and open boundary conditions

o) Finally, we thank referee 2 for the reference to arXiv:1906.07565. We added section 7 "The antiferromagnetic Ising model" and discussed the transformation from our results to the antiferromagnetic Ising model.

### Submission & Refereeing History

- Report 2 submitted on 2020-01-14 11:55 by
*Anonymous* - Report 1 submitted on 2019-12-29 04:18 by
*Anonymous*

## Reports on this Submission

### Anonymous Report 2 on 2020-1-14 Invited Report

### Strengths

As in my previous report.

### Weaknesses

The authors have improved the clarity of the paper by implementing the suggestions of the two referees.

### Report

I recommend acceptance in the present form. (The authors may however wish to mend a tiny BibTeX related issue regarding the title of reference [38].)

### Anonymous Report 1 on 2019-12-29 Invited Report

### Strengths

Spelled out in previous report.

### Weaknesses

I believe they have been addressed since the previous report. Still not easy to read, but that is to be expected, as the calculations are involved.

### Report

Publish this version, unless the other referee sees something still.

### Requested changes

No changes suggested