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 $d=3$.
تحميل البحث