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

Hendrik Hobrecht, Alfred Hucht

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

- doi: 10.21468/SciPostPhys.7.3.026
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### 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

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^{1 }Hendrik Hobrecht, -
^{1 }Fred Hucht