The Casimir Effect for the Bose-Gas in Slabs

– We study the Casimir effect for the perfect Bose-gase in the slab geometry for various boundary conditions. We show that the grand canonical potential per unit area at the bulk critical chemical potential μ = 0 has the standard asymptotic form with universal Casimir terms. In contrast to the well-known Casimir effect for the photon gas, see e.g. [1], this effect for the massive quantum particles (quantum gases) is much less explored. In the present letter we consider the case of the perfect Bose-gas. The perfect Bose-gas is the simplest quantum mechanical system showing the spontaneous breaking of a continuous symmetry (Bose-Einstein condensation) for which the Casimir effect should appear. To our knowledge, this has surprisingly not been studied before. We first recall the definition of the Casimir amplitude at critical points or for phases with broken symmetry [2] [3]. Let φd(T ) = lim L→∞ 1 L2 ΦL(T, d) (1) be a limiting thermodynamic potential per unit area in the 3-dimensional slab geometry ∞× ∞×d, where ΦL(T, d) is the finite-volume thermodynamic potential in the box Λ = L×L×d and T denotes the temperature. The finite size scaling analysis in the critical regime shows that large d-asymptotics of (1) has usually the form: φd(T ) = dφbulk(T ) + φsurf (T ) + ∆(T ) d2 + . . . . (2) Here φbulk(T ) is the potential density in thermodynamic limit and φsurf (T ) is the surface potential correction. The coefficient ∆(T ) in the third term in (2) is called the Casimir (∗) E-mail: [email protected] (∗∗) E-mail: [email protected]