Thermodynamic Casimir Effect in Films: the Exchange Cluster Algorithm

TL;DR

This paper investigates the thermodynamic Casimir effect in films with various boundary conditions using Monte Carlo simulations of the 3D Ising model, employing an efficient cluster algorithm to analyze finite size effects and universality.

Contribution

It introduces an efficient exchange cluster algorithm for studying the Casimir force in film geometries and analyzes finite size scaling near the phase transition.

Findings

Reduced statistical errors in Casimir force calculations.

Determined the inverse transition temperature for various film thicknesses.

Validated finite size scaling using the 2D Ising model's universal functions.

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. Bl\"ote, 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.