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

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 $2\times 2$ 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.

arXiv comment18 pages, title changed, accepted for publication

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Cited by in corpus (8)

• Critical Casimir Effect: Exact Results
• The square lattice Ising model on the rectangle I: Finite systems
• The square lattice Ising model on the rectangle II: Finite-size scaling limit
• Anisotropic scaling of the two-dimensional Ising model I: the torus
• Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields
• The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants
• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• The boundary disorder correlation for the Ising model on a cylinder