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.

18 pages, title changed, accepted for publication

References in corpus (7)

• Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations
• Critical Casimir forces in colloidal suspensions on chemically patterned surfaces
• Aspect-ratio dependence of thermodynamic Casimir forces
• Thermodynamic Casimir Forces between a Sphere and a Plate: Monte Carlo Simulation of a Spin Model
• Critical Casimir Interactions Between Spherical Particles in the Presence of the Bulk Ordering Fields
• Critical Casimir force in slab geometry with finite aspect ratio: analytic calculation above and below $T_c$
• Direct simulation of critical Casimir forces

Cited by in corpus (4)

• Critical Casimir Effect: Exact Results
• 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