Critical Casimir Effect: Exact Results

Abstract

If a material body is immersed into a fluctuating medium, its shape and the properties of its constituents modify the fluctuations in the surrounding medium. If in the same medium there is a second body, modifications of the fluctuation due to the first one influence the modifications due to the second one. This mutual influence results in a force between these bodies. If the fluctuating medium consists of the confined electromagnetic field in vacuum, one speaks of the quantum mechanical Casimir effect. In the case that the order parameter of material fields fluctuates - such as differences of number densities or concentrations - and that the corresponding fluctuations of the order parameter are long-ranged, one speaks of the critical Casimir effect. This holds, e.g., in the case of systems which undergo a second-order phase transition and which are thermodynamically located near the corresponding critical point, or for systems with a continuous symmetry exhibiting Goldstone mode excitations. Here we review the currently available exact results concerning the critical Casimir effect in systems encompassing the one-dimensional Ising, XY, and Heisenberg models, the two-dimensional Ising model, the Gaussian and the spherical models, as well as the mean field results for the Ising and the XY model. Special attention is paid to the influence of the boundary conditions on the behavior of the Casimir force.Comment: 218 pages, 67 figure