Critical Casimir forces for O(n) systems with long-range interaction in the spherical limit

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Critical Phenomena

Background:

• The Casimir effect describes a physical force arising from quantum fluctuations.
• Understanding thermodynamic Casimir forces is crucial for systems near critical points.
• Long-ranged interactions significantly influence system behavior at criticality.

Purpose of the Study:

• To present exact results for thermodynamic Casimir force and excess free energy.
• To investigate these quantities in a d-dimensional spherical model with power-law interactions.
• To analyze behavior near and above/below the critical temperature in a film geometry.

Main Methods:

• Utilized the d-dimensional spherical model with power-law decaying interactions (r^{-d-sigma}).
• Analyzed film geometry under periodic boundary conditions.
• Derived and investigated the universal finite-size scaling function for the Casimir force.

Main Results:

• Derived exact results for thermodynamic Casimir force and excess free energy.
• The Casimir force is shown to be universally attractive for sigma >= 1.
• Identified distinct force dependencies on film thickness (L) based on temperature relative to T(c).

Conclusions:

• The excess free energy is always negative and increases with temperature and magnetic field.
• The Casimir force is always attractive for sigma >= 1, indicating surface attraction.
• The force exhibits complex behavior at T(c), depending on sigma and magnetic field.