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

Abstract

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 ${\varphi }^4$ 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.

arXiv commentComment on arXiv:1210.1452, see also arXiv:1402.3510, accepted

References in corpus (4)

• Large-$n$ approach to thermodynamic Casimir effects in slabs with free surfaces
• Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces
• Casimir force in the O(n -> infinity) model with free boundary conditions
• The O(n) ${\varphi }^4$ model with free surfaces in the large-$n$ limit: Some exact results for boundary critical behaviour, fluctuation-induced forces and distant-wall corrections

Cited by in corpus (9)

• Critical Casimir Effect: Exact Results
• Fluctuation-induced forces in confined ideal and imperfect Bose gases
• Crossover from low-temperature to high-temperature fluctuations. I. Thermodynamic Casimir forces of isotropic systems
• Crossover from low-temperature to high-temperature fluctuations. II. Nonuniversal thermodynamic Casimir forces of anisotropic systems
• Line contribution to the critical Casimir force between a homogeneous and a chemically stepped surface
• Inverse scattering-theory approach to the exact large-$n$ solutions of $O(n)$ ${\varphi }^4$ models on films and semi-infinite systems bounded by free surfaces
• Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials
• Reply to Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• The three-dimensional $O(n)$ ${\varphi }^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit $n\to \infty$