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

H. W. Diehl, Daniel Grüneberg, Martin Hasenbusch, Alfred Hucht, Sergei B. Rutkevich, Felix M. Schmidt

Published 2014-05-22, updated 2014-12-10Version 2

In a recent paper [D. Dantchev, J. Bergnoff, and J. Rudnick, Phys. Rev. E 89, 042116 (2014)] the problem of the Casimir force in the $O(n)$ model on a slab with free boundary conditions, investigated earlier by us [EPL 100, 10004 (2012)], is reconsidered using a mean spherical model with separate constraints for each layer. The authors (i) question the applicability of the Ginzburg-Landau-Wilson approach to the low-temperature regime, arguing for the superiority of their model compared to the family of $\phi^4$ models A and B whose numerically exact solutions we determined both for values of the coupling constant $0<g<\infty$ and $g=\infty$. They (ii) report consistency of their results with ours in the critical region and a strong manifestation of universality, but (iii) point out discrepancies with our results in the region below $T_{\mathrm{c}}$. We show here that (i) is unjustified and prove that our model B with $g=\infty$ is identical to their spherical model. Hence evidence for the reported universality is already contained in our work. Moreover, the results we determined for anyone of the models A and B for various thicknesses $L$ are all numerically exact. (iii) is due to their misinterpretation of our results for the scaling limit. We also show that their low-temperature expansion, which does not hold inside the scaling regime, is limited to temperatures lower than they anticipated.