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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 $Ltimes M$ 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 $2times2$ 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 $rhoto 0$. Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.
Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling
We have dramatically extended the zero field susceptibility series at both high and low temperature of the Ising model on the triangular and honeycomb lattices, and used these data and newly available further terms for the square lattice to calculate
We present general arguments and construct a stress tensor operator for finite lattice spin models. The average value of this operator gives the Casimir force of the system close to the bulk critical temperature $T_c$. We verify our arguments via exa
The critical Casimir force (CCF) arises from confining fluctuations in a critical fluid and thus it is a fluctuating quantity itself. While the mean CCF is universal, its (static) variance has previously been found to depend on the microscopic detail
We present results of a Monte Carlo study for the ferromagnetic Ising model with long range interactions in two dimensions. This model has been simulated for a large range of interaction parameter $sigma$ and for large sizes. We observe that the resu