Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios

TL;DR

This paper develops a systematic method to compute universal scaling functions for the critical Casimir force in the 2D Ising model with various boundary conditions, confirming known results and extending to new geometries.

Contribution

It introduces a transfer matrix approach to calculate the critical Casimir scaling functions for different boundary conditions in the 2D Ising model.

Findings

Reproduces known scaling functions for thin films as aspect ratio approaches zero.

Provides detailed calculations for the cylinder geometry with open boundary conditions.

Confirms conformal field theory predictions at criticality for the cylinder geometry.

Abstract

We present a systematic method to calculate the universal scaling functions for the critical Casimir force and the according potential of the two-dimensional Ising model with various boundary conditions. Therefore we start with the dimer representation of the corresponding partition function $Z$ on an $L\times M$ square lattice, wrapped around a torus with aspect ratio $\rho=L/M$. By assuming periodic boundary conditions and translational invariance in at least one direction, we systematically reduce the problem to a $2\times2$ transfer matrix representation. For the torus we first reproduce the results by Kaufman and then give a detailed calculation of the scaling functions. Afterwards we present the calculation for the cylinder with open boundary conditions. All scaling functions are given in form of combinations of infinite products and integrals. Our results reproduce the known scaling functions in the limit of thin films $\rho\to 0$. Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.