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

Abstract

At variance with the authors' statement [L. Pálová, 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 oneloop approach, is consistent with some previous analytical results obtained in the framework of the εexpansion 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.

Quantum critical scaling and the Gross-Neveu model in 2+1 dimensions

“…In analogy to the critical Casimir effect due to spatial confinement, we consider the critical behavior of quantum systems in the context of the critical Casimir effect, defining the temporal Casimir effect [9,10] also dubbed Casimir effect in time [11], where the confinement is caused by the boundary conditions along the finite imaginary time direction. The singular part of the free energy of a ddimensional system in the vicinity of its quantum critical point, with hyperscaling satisfied, is expressed as…”