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
1
2
σ<d<
3
2
σ
, where
0<σ≤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
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
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
c
~
=0.606
.