Reply to Comment on "Casimir force in the $O(n\to\infty)$ model with free boundary conditions"

TL;DR

This paper defends a previous study on the Casimir force in the $O(n o\infty)$ model with free boundary conditions, emphasizing the use of a single model across all temperatures to clarify crossover effects.

Contribution

It demonstrates the advantages of using a single model over a range of temperatures to study the Casimir force, contrasting with multiple models used in prior work.

Findings

The single model approach clarifies the crossover from critical to Goldstone mode effects.

The low-temperature analytical expansion remains accurate as system size increases.

The model's validity extends closer to the critical temperature without entering the critical region.

Abstract

The proceeding comment raises a few points concerning our paper Dantchev \textit{et al.}, Phys. Rev. E. {\bf 89}, 042116 (2014). In this reply we stress that while Refs. Diehl \textit{et al.} EPL {\bf 100}, 10004 (2012) and Phys. Rev. E. {\bf 89}, 062123 (2014) use three different models to study the the Casimir force for the $O(n \rightarrow \infty)$ model with free boundary conditions we study a single model over the entire range of temperatures, from above the bulk critical temperature, $T_c$, to absolute temperatures down to $T=0$. The use of a single model renders more transparent the crossover from effects dominated by critical fluctuations in the vicinity of the bulk transition temperature to effects controlled by Goldstone modes at low temperatures. Contrary to the assertion in the comment, we make no claim for the superiority of our model over any of those considered by Diehl \textit{et al}. We also present additional evidence supporting our conclusion in Dantchev \textit{et al.}, Phys. Rev. E. {\bf 89}, 042116 (2014) that the temperature range in which our low-temperature analytical expansion for the Casimir force increases as $L$ grows and remains accurate for values of the ratio $T/T_c$ that become closer and closer to unity, while $T$ remains well outside of the critical region.