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

TL;DR

This paper provides a comprehensive analysis of the finite-size effects and anisotropic scaling behavior of the two-dimensional Ising model on a torus, including partition functions, free energy, and universal scaling functions.

Contribution

It extends finite-size scaling theory to anisotropic systems and details the impact of boundary conditions and aspect ratio on the Ising model's thermodynamic properties.

Findings

Derived explicit partition functions for various boundary conditions.

Identified how anisotropy influences finite-size scaling variables.

Showed the emergence of surface tension effects with antiperiodic boundaries.

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.