Critical Casimir Effect: Exact Results

Abstract

In any medium there are fluctuations due to temperature or due to the quantum nature of its constituents. If a material body is immersed into such a 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 excitations of the medium, which mediate the effective interaction between the bodies, are massless, this force is longranged and nowadays known as a Casimir force. 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. We present results both for the case of classical critical fluctuations if the system possesses a critical point at a non-zero temperature, as well as the case of quantum systems undergoing a continuous phase transition at zero temperature as function of certain parameters. As confinements we consider the film, the sphere -plane, and the sphere -sphere geometries. We discuss systems governed by short-ranged, by subleading long-ranged (i.e., of the van der Waals type), and by leading long-ranged interactions. In order to put the critical Casimir effect into the proper context and in order to make the review as self-contained as possible, basic facts about the theory of phase transitions, the theory of critical phenomena in classical and quantum systems, and finite-size scaling theory are recalled. Whenever possible, a discussion of the relevance of the exact results towards an understanding of available experiments is presented. Hints about eventually applying the results towards their potential use in devices are also given.

Variance of the Casimir force in an ideal Bose gas

“…= 0, see equations ( 22) and (23), this leads to the following expression for the amplitude Δ var (T , μ, D) in equation ( 25)…”

Tracer particle in a confined correlated medium: an adiabatic elimination method

“…Moreover, recent advances in experimental technology have made it possible to measure the critical Casimir forces acting on colloidal particles immersed in near-critical fluid media [34][35][36]. The properties of these forces, which are the classical thermal counterpart of the celebrated Casimir force in electromagnetism [37], are relatively well understood in equilibrium [38][39][40][41][42][43], while much remains to be unveiled about their non-equilibrium properties. Simple toy models have thus been devised in order to investigate the coupled out of equilibrium dynamics of one or a few particles in contact with near-critical media [14,15,17,[21][22][23]44].…”

Phase transition of photons and gravitons in a Casimir box

“…At low temperature, this is accompanied by a quantum phase transition of the same type as that studied in finite size systems [8], see e.g. [9,10] for reviews. In other words, we work out the quantum critical Casimir effect for photons and gravitons.…”