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

TL;DR

This paper investigates how surfaces and boundary fields influence the anisotropic 2D Ising model, confirming theoretical predictions and extending results to antiferromagnetic cases in finite and scaling limits.

Contribution

It provides explicit calculations of boundary effects on the anisotropic Ising model, validating scaling and conformal theories, and extends findings to antiferromagnetic systems.

Findings

Anisotropic couplings affect scaling functions via a generalized aspect ratio.

Open and staggered boundary conditions become equivalent in the scaling limit.

Surface tension emerges due to antiperiodic boundaries with symmetry-breaking fields.

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.