Comment on "Quantum critical paraelectrics and the Casimir effect in time"

TL;DR

This paper clarifies the behavior of the universal scaling amplitude in phonon dispersion near quantum critical points, confirming previous analytical results and numerically analyzing the Casimir amplitude across dimensions.

Contribution

It demonstrates the consistency of a self--consistent one--loop approach with earlier analytical and exact results, and numerically investigates the Casimir amplitude's behavior with dimensionality.

Findings

The scaling amplitude aligns with previous analytical and exact results.

The temporal Casimir amplitude peaks at approximately d=2.9144.

Numerical expansion confirms the Casimir amplitude behavior near d=3.

Abstract

At variance with the authors' statement [L. P\'{a}lov\'{a}, P. Chandra and P. Coleman, Phys. Rev. B 79, 075101 (2009)], we show that the behavior of the universal scaling amplitude of the gap function in the phonon dispersion relation as a function of the dimensionality $d$, obtained within a self--consistent one--loop approach, is consistent with some previous analytical results obtained in the framework of the $\epsilon$--expansion in conjunction with the field theoretic renormalization group method [S. Sachdev, Phys. Rev. B 55, 142 (1997)] and the exact calculations corresponding to the spherical limit i.e. infinite number $N$ of the components of the order parameter [H. Chamati. and N. S. Tonchev, J. Phys. A: Math. Gen. 33, 873 (2000)]. Furthermore we determine numerically the behavior of the "temporal" Casimir amplitude as a function of the dimensionality $d$ between the lower and upper critical dimension and found a maximum at $d=2.9144$. This is confirmed via an expansion near the upper dimension $d=3$.