Thermodynamic Casimir Effect in Films: the Exchange Cluster Algorithm

Abstract

We study the thermodynamic Casimir force for films with various types of boundary conditions and the bulk universality class of the three-dimensional Ising model. To this end we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. In particular, we employ the exchange or geometric cluster cluster algorithm [J.R. Heringa and H. W. J. Blöte, Phys. Rev. E 57, 4976 (1998)]. In a previous work we demonstrated that this algorithm allows to compute the thermodynamic Casimir force for the plate-sphere geometry efficiently. It turns out that also for the film geometry a substantial reduction of the statistical error can achieved. Concerning physics, we focus on (O,O) boundary conditions, where O denotes the ordinary surface transition. These are implemented by free boundary conditions on both sides of the film. Films with such boundary conditions undergo a phase transition in the universality class of the two-dimensional Ising model. We determine the inverse transition temperature for a large range of thicknesses L_0 of the film and study the scaling of this temperature with L_0. In the neighborhood of the transition, the thermodynamic Casimir force is affected by finite size effects, where finite size refers to a finite transversal extension L of the film. We demonstrate that these finite size effects can be computed by using the universal finite size scaling function of the free energy of the two-dimensional Ising model.

48 pages, 20 figures

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• Critical and near critical phase behaviour and interplay between the thermodynamic Casimir and van der Waals forces in confined non-polar fluid medium with competing surface and substrate potentials
• A variance reduced estimator of the connected two-point function in the presence of a broken Z_2 symmetry
• Thermodynamic Casimir forces in strongly anisotropic systems within the $N\to \infty$ class
• Finite-size scaling of the magnetization probability density for the critical Ising model in slab geometry
• Spin models in three dimensions: Adaptive lattice spacing
• Effective binding potential from Casimir interactions: the case of the Bose gas
• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• Casimir effect in critical $\mathrm{O}(N)$ models from non-equilibrium Monte Carlo simulations
• The interface tension in the improved Blume-Capel model