Casimir amplitudes in a quantum spherical model with long-range interaction

A d-dimensional quantum model system confined to a general hypercubical geometry with linear spatial size L and "temporal size"1/T ( T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions $\frac 12\sigma < d< \frac 32\sigma $, where $0< \sigma \leq 2$ is a parameter controlling the decay of the long-range interaction, the free energy and the Casimir amplitudes are given. We have proven that, if $d=\sigma$, the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state, is $\Delta =-16\zeta(3)/[5\sigma(4\pi )^{\sigma/2}\Gamma (\sigma /2)]$. The last implies that the universal constant $\tilde{c}=4/5$ of the model remains the same for both short, as well as long-range interactions, if one takes the normalization factor for the Gaussian model to be such that $\tilde{c}=1$. This is a generalization to the case of long-range interaction of the well-known result due to Sachdev. That constant differs from the corresponding one characterizing the leading finite-size corrections at zero temperature which for $d=\sigma=1$ is $\tilde c=0.606$