Systems with an O(n) symmetrical Hamiltonian are considered in a d-dimensional slab geometry of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L → ∞. The effective forces induced by thermal fluctuations at and above the bulk critical temperature Tc,∞ (thermodynamic Casimir effect) are investigated below the upper critical dimension d * = 4 by means of field-theoretic renormalization group methods for the case of periodic and special-special boundary conditions, where the latter correspond to the critical enhancement of the surface interactions on both boundary planes. As shown previously [Europhys. Lett. 75, 241 (2006)], the zero modes that are present in Landau theory at Tc,∞ make conventional RG-improved perturbation theory in 4−ǫ dimensions ill-defined. The revised expansion introduced there is utilized to compute the scaling functions of the excess free energy and the Casimir force for temperatures T ≥ Tc,∞ as functions of L ≡ L/ξ∞, where ξ∞ is the bulk correlation length. Scaling functions of the L-dependent residual free energy per area are obtained whose L → 0 limits are in conformity with previous results for the Casimir amplitudes ∆C to O(ǫ 3/2 ) and display a more reasonable small-L behavior inasmuch as they approach the critical value ∆C monotonically as L → 0. Extrapolations to d = 3 for the Ising case n = 1 with periodic boundary conditions are in fair agreement with Monte Carlo results. In the case of special-special boundary conditions, extrapolations to d = 3 are hampered by the fact that the one-loop result for the inverse finite-size susceptibility becomes negative for some values of L when ǫ 0.83.

The classical n-vector ϕ{4} model with O(n) symmetrical Hamiltonian H is considered in a ∞{2}×L slab geometry bounded by a pair of parallel free surface planes at separation L. Standard quadratic boundary terms implying Robin boundary conditions are included in H. The temperature-dependent scaling functions of the excess free energy and the thermodynamic Casimir force are computed in the large-n limit for temperatures T at, above, and below the bulk critical temperature T_{c}. Their n=∞ limits can be expressed exactly in terms of the spectrum and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. This equation is solved by numerical means for two distinct discretized versions of the model: in the first ("model A"), only the coordinate z across the slab is discretized and the integrations over momenta conjugate to the lateral coordinates are regularized dimensionally; in the second ("model B"), a simple cubic lattice with periodic boundary conditions along the lateral directions is used. Renormalization-group ideas are invoked to show that, in addition to corrections to scaling ∝L{-1}, anomalous ones ∝L{-1}lnL should occur. They can be considerably decreased by taking an appropriate g→∞ (T_{c}→∞) limit of the ϕ{4} interaction constant g. Depending on the model A or B, they can be absorbed completely or to a large extent in an effective thickness L_{eff}=L+δL. Excellent data collapses and consistent high-precision results for both models are obtained. The approach to the low-temperature Goldstone values of the scaling functions is shown to involve logarithmic anomalies. The scaling functions exhibit all qualitative features seen in experiments on the thinning of wetting layers of {4}He and Monte Carlo simulations of XY models, including a pronounced minimum of the Casimir force below T_{c}. The results are in conformity with various analytically known exact properties of the scaling functions.

We consider the three-dimensional Ising model in a L(⊥)×L(∥)×L(∥) cuboid geometry with a finite aspect ratio ρ=L(⊥)/L(∥) and periodic boundary conditions along all directions. For this model the finite-size scaling functions of the excess free energy and thermodynamic Casimir force are evaluated numerically by means of Monte Carlo simulations. The Monte Carlo results compare well with recent field theoretical results for the Ising universality class at temperatures above and slightly below the bulk critical temperature T(c). Furthermore, the excess free energy and Casimir force scaling functions of the two-dimensional Ising model are calculated exactly for arbitrary ρ and compared to the three-dimensional case. We give a general argument that the Casimir force vanishes at the critical point for ρ=1 and becomes repulsive in periodic systems for ρ>1.

Abstract. -Systems described by n-component φ 4 models in a ∞ d−1 × L slab geometry of finite thickness L are considered at and above their bulk critical temperature Tc,∞. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces ("thermodynamic Casimir effect") in d = 4 − ǫ bulk dimensions is re-examined. It is found to be ill-defined beyond two-loop order because of infrared singularities when the boundary conditions are such that the free propagator in slab geometry involves a zero-energy mode at bulk criticality. This applies to periodic boundary conditions and the special-special ones corresponding to the critical enhancement of the surface interactions on both confining plates. The field theory is reorganized such that a small-ǫ expansion results which remains well behaved down to Tc,∞. The leading contributions to the critical Casimir amplitudes ∆per and ∆sp,sp beyond two-loop order are ∼ (u * ) (3−ǫ)/2 , where u * = O(ǫ) is the value of the renormalized φ 4 coupling at the infrared-stable fixed point. Besides integer powers of ǫ, the small-ǫ expansions of these amplitudes involve fractional powers ǫ k/2 , with k ≥ 3, and powers of ln ǫ. Explicit results to order ǫ Fluctuations associated with long wave-length, low-energy excitations play a crucial role in determining the physical properties of many macroscopic systems. When such fluctuations are confined by boundaries, walls, or size restrictions along one or several axes, important effective forces may result. In those cases where the continuous mode spectrum that emerges as the system becomes macroscopic in all directions is not separated from zero energy by a gap, these fluctuation-induced forces are longranged, decaying algebraically as a function of the relevant confinement length L (separation of walls, thickness of the system, etc).A prominent example of such forces are the Casimir forces [1] induced by vacuum fluctuations of the electromagnetic field between two metallic bodies a distance L apart [2][3][4]. Analogous long-range effective forces occur in condensed matter systems as the result of either (i) thermal fluctuations at continuous phase transitions or else (ii) Goldstone modes and similar "massless" excitations [5][6][7][8][9][10][11][12]. In particular the former ones, frequently called "critical Casimir forces," have attracted considerable theoretical and experimental attention recently. Beginning with the seminal paper by Fisher and de Gennes [5], they have been studied theoretically for more than a decade using renormalization group (RG) [6][7][8][9][10] and conformal field theory methods [13], exact solutions of models [12], as well as Monte c EDP Sciences

-The limit n → ∞ of the classical O(n) φ 4 model on a 3d film with free surfaces is studied. Its exact solution involves a selfconsistent 1d Schrödinger equation, which is solved numerically for a partially discretized as well as for a fully discrete lattice model. Extremely precise results are obtained for the scaled Casimir force at all temperatures. Obtained via a single framework, they exhibit all relevant qualitative features of the thermodynamic Casimir force known from wetting experiments on 4 He and Monte Carlo simulations, including a pronounced minimum below the bulk critical point.A celebrated example of fluctuation-induced forces is the Casimir force between two metallic, grounded plates in vacuum [1].1 Such forces caused by the confinement of quantum electrodynamics (QED) vacuum fluctuations of the electromagnetic fields are expected to have considerable technological relevance. This has made them the focus of much ongoing research activity. During the past two decades, it has become increasingly clear that a wealth of similarly interesting classical analogs of such effective forces, induced by thermal rather than quantum fluctuations, exist [3].2 Two important classes of such "thermodynamic Casimir forces" 3 are forces induced by fluctuations in nearly (multi)critical media between immersed macroscopic bodies or boundaries, and forces due to confined Goldstone modes [6]. Clear experimental evidence for the existence of such thermodynamic Casimir 1 For a review of the Casimir effect in QED and an extensive lists of references, see [2] 2 For reviews of the thermodynamic Casimir effect and extensive lists of references, see [4] 3 Following established conventions we use the term "thermodynamic Casimir forces" for forces induced by thermal fluctuations, in particular, also for near-critical Casimir forces, reserving the name critical Casimir forces to those where the medium is at a critical point. This topic must not be confused with those of thermal effects on QED Casimir forces and thermal Casimir-Polder forces, which are less universal since material properties of the media and confining objects matter; see, e.g., [5] forces was provided first indirectly by measurements of the thinning of 4 He wetting films at the λ-point as the temperature T is lowered below the bulk critical temperature T c [7]. Subsequently, direct measurements of the thermodynamic Casimir force on colloidal particles in binary liquids near the consolute point could be achieved [8].Despite obvious analogies, crucial qualitative differences between thermodynamic and QED Casimir forces exist. First, the latter usually can be studied in terms of effective free field theories in confined geometries where the interaction of the electromagnetic field with the material boundaries is taken into account through boundary conditions. By contrast, investigations of thermodynamic Casimir forces at (multi)critical points necessarily involve interacting field theories. Second, whereas electromagnetic fields average to zero in the groun...