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

Abstract

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 > Tc) the Casimir force decays exponentially as a function of D, with the bulk correlation length determining the relevant length scale. For T = Tc and for T < Tc its decay is algebraic. The magnitude of the Casimir forces at Tc and for T < Tc 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 monotonic at any d∈(2,4) and becomes constant for d > 4 and T ≤ Tc.