The imperfect Bose gas in d dimensions: critical behavior and Casimir forces

We consider the d-dimensional imperfect (mean-field) Bose gas confined in a slit-like geometry and subject to periodic boundary conditions. Within an exact analytical treatment we first extract the bulk critical properties of the system at Bose-Einstein condensation and identify the bulk universality class to be the one of the classical d-dimensional spherical model. Subsequently we consider finite slit width D and analyze the excess surface free energy and the related Casimir force acting between the slit boundaries. Above the bulk condensation temperature (T>T_c) the Casimir force decays exponentially as a function of D with the bulk correlation length determining the relevant length scale. For T=T_c and for T<T_c its decay is algebraic. The magnitude of the Casimir forces at T_c and for T<T_c is governed by the universal Casimir amplitudes. We extract the relevant values for different d and compute the scaling functions describing the crossover between the critical and low-temperature asymptotics of the Casimir force. The scaling function is monotonous at any d\in (2,4) and becomes constant for d>4 and T\leq T_c.