Critical free energy and Casimir forces in rectangular geometries

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Critical Phenomena

Background:

• The thermodynamic Casimir effect describes forces arising from boundary conditions in systems near criticality.
• Understanding finite-size scaling and geometric crossovers is crucial for characterizing critical behavior.

Purpose of the Study:

• To investigate the critical behavior of the free energy and thermodynamic Casimir force in diverse geometries.
• To analyze the geometric crossover from film to cubic to cylindrical shapes.
• To compare theoretical predictions with experimental and simulation data.

Main Methods:

• Utilizing the O(n) symmetric ϕ4 lattice model with periodic boundary conditions.
• Deriving exact results in the large-n limit for geometric crossovers.
• Employing three perturbation approaches in the minimal renormalization scheme for n=1.
• Analyzing finite-size scaling functions.

Main Results:

• Exact results in the large-n limit describe geometric crossovers in film, cubic, and cylindrical geometries.
• Predictions for the critical Casimir force at bulk T(c): attractive for slabs (ρ<1), repulsive for rods (ρ>1), and zero for cubes (ρ=1).
• Finite-size scaling functions show good agreement with Monte Carlo data for the 3D Ising model.

Conclusions:

• The study provides a comprehensive analysis of the thermodynamic Casimir force across different geometries and dimensions.
• Theoretical predictions are validated by experimental and simulation data, confirming the model's accuracy.
• The findings offer insights into critical phenomena and finite-size effects in diverse physical systems.