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&#39;&#39; $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 $\frac 12σ<d<\frac 32σ$, where $0<σ\leq 2$ 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 $d=σ$, the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state, is $Δ=-16ζ(3)/[5σ(4π)^{σ/2}Γ(σ/2)]$. 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 $d=σ=1$ is $\tilde c=0.606$.