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

Abstract

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 $\frac{1}{2}\sigma <d<\frac{3}{2}\sigma$, where $0<\sigma \le 2$ 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 $\tilde{c}=0.606$.

10 pages latex, no figures, to appear in EPJB (2000)