Casimir amplitudes in a quantum 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 ∆ = −16ζ(3)/[5σ(4π)Γ (σ/2)]. The last implies that the universal constant c̃ = 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 c̃ = 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 c̃ = 0.606.