SciPost Phys. 8, 032 (2020) ·
published 2 March 2020

· pdf
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
finitesize 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.
SciPost Phys. 7, 026 (2019) ·
published 2 September 2019

· pdf
We present detailed calculations for the partition function and the free
energy of the finite twodimensional 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 finitesize
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 finitesize 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
finitesize scaling theory to the anisotropic case and show how this anisotropy
can be absorbed into suitable scaling variables.
Dr Hucht: "We thank Prof. Perk for his de..."
in Report on Anisotropic scaling of the twodimensional Ising model I: The torus