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