Casimir force in the O(n -> infinity) model with free boundary conditions

TL;DR

This paper investigates the temperature-dependent Casimir force in a film geometry described by the infinite-component limit of the O(n) model, covering all temperature regimes including critical and low-temperature phases.

Contribution

It provides a comprehensive analysis of the Casimir force across all temperatures for the O(n) model with free boundary conditions, extending previous studies to include the entire temperature range.

Findings

Force behavior near critical temperature is dominated by critical fluctuations.

At low temperatures, Goldstone modes significantly influence the force.

Results qualitatively resemble experimental data for ^4He films.

Abstract

We present results for the temperature behavior of the Casimir force for a system with a film geometry with thickness $L$ subject to free boundary conditions and described by the $n\to\infty$ limit of the $O(n)$ model. These results extend over all temperatures, including the critical regime near the bulk critical temperature $T_c$, where the critical fluctuations determine the behavior of the force, and temperatures well below it, where its behavior is dictated by the Goldstone's modes contributions. The temperature behavior when the absolute temperature, $T$, is a finite distance below $T_c$, up to a logarithmic-in-$L$ proximity of the bulk critical temperature, is obtained both analytically and numerically; the critical behavior follows from numerics. The results resemble - but do not duplicate - the experimental curve behavior for the force obtained for $^4$He films.