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 study the excitation energy spectrum in the S = 1/2 ferromagnetic Ising spin chain with the easy axis z in a magnetic field h = {h x , 0, h z }. According to Wu and McCoy's scenario of weak confinement, the fermionic spinon excitations (kinks), being free at h z = 0 in the ordered phase, are coupled into bosonic bound states at arbitrary small h z > 0. We calculate the energy spectrum of such excitations in the leading order in small h z , using different perturbative methods developed for the similar problem in the Ising field theory.

-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...

In the presence of a small magnetic field h, the elementary excitations in the scaling two-dimensional Ising model are studied perturbatively in in the ferromagnetic phase. For excitations with large numbers n, the mass spectrum is obtained in the first order in h. The decay widths of excitations with energies above the stability threshold are calculated in the leading h3 order.