Comment on “Casimir force in the $\mathrm{O}\left(\mathit{n}\to \infty \right)$ model with free boundary conditions”

In a recent paper by D. Dantchev, J. Bergknoff, and J. Rudnick [Phys. Rev. E 89, 042116 (2014)], the problem of the Casimir force in the $\mathrm{O}\left(n\right)$ model on a slab with free boundary conditions, investigated earlier by us [Europhys. Lett. 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 for $g=\infty$. 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 $T_c$. Here we refute (i) 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 paper. 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.

Figure 1

Asymptotic behavior of the Casimir force scaling function $\vartheta \left(x\right)$ for $x\to -\infty$ (thick dot-dashed line), together with the uncorrected data ${\vartheta }_L\left(x\right)$ of DBR's model for $L=9,17,33,...,1025$ (solid lines). For $x\lesssim -L$ the results deviate from the scaling function and approach the expansion Eqs. (3.21)– (3.26) of Ref. [1] shown as dotted lines, cf. Ref. [3, Fig. 6]. This expansion never captures the correct asymptotic behavior of the scaling function. The inset shows the crossover from the scaling regime to the low-temperature regime at $t=x/L\approx -1$.