H. W. Diehl

Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

Systems described by an O(n) symmetrical varphi;{4} Hamiltonian are considered in a d-dimensional film geometry at their bulk critical points. The critical Casimir forces between the films boundary planes B_{j}, j=1,2, are investigated as functions of film thickness L for generic symmetry-preserving boundary conditions partial differential_{n}phi=c[over composite function]_{j}phi. The L-dependent part of the reduced excess free energy per cross-sectional area takes the scaling form f_{res} approximately D(c_{1}L;{Phi/nu},c_{2}L;{Phi/nu})/L;{d-1} when d<4, where c_{i} are scaling fields associated with the variables c[over composite function]_{i} and Phi is a surface crossover exponent. Explicit two-loop renormalization group results for the function D(c_{1},c_{2}) at d=4- dimensions are presented. These show that (i) the Casimir force can have either sign, depending on c_{1} and c_{2}, and (ii) for appropriate choices of the enhancements c[over composite function]_{j}, crossovers from attraction to repulsion and vice versa occur as L increases.

Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces

The limit n???? of the classical O(n) ?4 model on a 3d film with free surfaces is studied. Its exact solution involves a self-consistent 1d Schr?dinger 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 4He and Monte Carlo simulations, including a pronounced minimum below the bulk critical point.

The critical Casimir effect in films for generic non-symmetry-breaking boundary conditions

Systems described by an O(n) symmetrical ?4 Hamiltonian are considered in a d-dimensional film geometry at their bulk critical points. A detailed renormalization group (RG) study of the critical Casimir forces induced between the films boundary planes , by thermal fluctuations is presented for the case where the O(n) symmetry remains unbroken by the surfaces. The boundary planes are assumed to cause short-range disturbances of the interactions that can be modelled by standard surface contributions ??2 corresponding to subcritical or critical enhancement of the surface interactions. This translates into mesoscopic boundary conditions of the generic symmetry-preserving Robin type . RG-improved perturbation theory and Abel?Plana techniques are used to compute the L-dependent part fres of the reduced excess free energy per film area A???? to two-loop order. When d?<?4, it takes the scaling form fres???D(c1L?/?,c2L?/?)/Ld?1 as L????, where ci are scaling fields associated with the surface-enhancement variables c? i, while ? is a standard surface crossover exponent. The scaling function D(c1,c2) and its analogue for the Casimir force are determined via expansion in ??=?4???d and extrapolated to d?=?3 dimensions. In the special case c1?=?c2?=?0, the expansion becomes fractional. Consistency with the known fractional expansions of D(0,0) and to order ?3/2 is achieved by appropriate reorganization of RG-improved perturbation theory. For appropriate choices of c1 and c2, the Casimir forces can have either sign. Furthermore, crossovers from attraction to repulsion and vice versa may occur as L increases.

Critical Casimir amplitudes for n-component ϕ4 models with O(n)-symmetry breaking quadratic boundary terms

Abstract Euclidean n -component ϕ 4 theories whose Hamiltonians are O ( n ) symmetric except for quadratic symmetry breaking boundary terms are studied in the film geometry R d − 1 × [ 0 , L ] . The boundary terms imply the Robin boundary conditions ∂ n ϕ α = c ˚ α ( j ) ϕ α at the boundary planes B j = 1 at z = 0 and B j = 2 at z = L . Particular attention is paid to the cases in which m j of the n variables c ˚ α ( j ) associated with plane B j take the special value c ˚ m j - sp corresponding to critical enhancement while the remaining ones are larger and hence subcritically enhanced. Under these conditions, the semi-infinite system with boundary plane B j has a multicritical surface–bulk point, called m j -special, at which an O ( m j ) symmetric critical surface phase coexists with the O ( n ) symmetric bulk phase, provided d is sufficiently large. The L -dependent part of the reduced free energy per cross-section area behaves asymptotically as Δ C / L d − 1 as L → ∞ at the bulk critical point. The Casimir amplitudes Δ C are determined for small ϵ = 4 − d in the general case where m c , c components ϕ α are critically enhanced at both boundary planes, m c , D + m D , c components are enhanced at one plane but satisfy asymptotic Dirichlet boundary conditions at the respective other, and the remaining m D , D components satisfy asymptotic Dirichlet boundary conditions at both B j . Whenever m c , c > 0 , the corresponding small-ϵ expansions involve, besides integer powers of ϵ, also fractional powers ϵ k / 2 with k ⩾ 3 modulo powers of logarithms. Results to order ϵ 3 / 2 are given for general values of m c , c , m c , D + m D , c , and m D , D . These are used to estimate the Casimir amplitudes Δ C of the three-dimensional Heisenberg systems with surface spin anisotropies for the cases with ( m c , c , m c , D + m D , c ) = ( 1 , 0 ) , ( 0 , 1 ) , and ( 1 , 1 ) .

Inverse-scattering-theory approach to the exact n→∞ solutions of O(n) ϕ

The O(n) ϕ(4) model on a strip bounded by a pair of planar free surfaces at separation L can be solved exactly in the large-n limit in terms of the eigenvalues and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. The scaling limit of a continuum version of this model is considered. It is shown that the self-consistent potential can be eliminated in favor of scattering data by means of appropriately extended methods of inverse scattering theory. The scattering data (Jost function) associated with the self-consistent potential are determined for the L=∞ semi-infinite case in the scaling regime for all values of the temperature scaling field t=(T-T(c))/T(c) above and below the bulk critical temperature T(c). These results are used in conjunction with semiclassical and boundary-operator expansions and a trace formula to derive exact analytical results for a number of quantities such as two-point functions, universal amplitudes of two excess surface quantities, the universal amplitude difference associated with the thermal singularity of the surface free energy, and potential coefficients. The asymptotic behaviors of the scaled eigenenergies and eigenfunctions of the self-consistent Schrödinger equation as function of x=t(L/ξ(+))(1/ν) are determined for x→-∞. In addition, the asymptotic x→-∞ forms of the universal finite-size scaling functions Θ(x) and ϑ(x) of the residual free energy and the Casimir force are computed exactly to order 1/x, including their x(-1)ln|x| anomalies.

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In a recent paper by D. Dantchev, J. Bergknoff, and J. Rudnick [Phys. Rev. E 89, 042116 (2014)], the problem of the Casimir force in the O(n) model on a slab with free boundary conditions, investigated earlier by us [Europhys. Lett. 100, 10004 (2012)], is reconsidered using a mean-spherical model with separate constraints for each layer. The authors (i) question the applicability of the Ginzburg-Landau-Wilson approach to the low-temperature regime, arguing for the superiority of their model compared to the family of ϕ(4) models A and B whose numerically exact solutions we determined both for values of the coupling constant 0<g<∞ and for g=∞. They (ii) report consistency of their results with ours in the critical region and a strong manifestation of universality but (iii) point out discrepancies with our results in the region below T(c). Here we refute (i) and prove that our model B with g=∞ is identical to their spherical model. Hence evidence for the reported universality is already contained in our paper. Moreover, the results we determined for anyone of the models A and B for various thicknesses L are all numerically exact. (iii) is due to their misinterpretation of our results for the scaling limit. We also show that their low-temperature expansion, which does not hold inside the scaling regime, is limited to temperatures lower than they anticipated.

models on films and semi-infinite systems bounded by free surfaces

Inverse scattering theory is extended to one-dimensional Schr?dinger problems with near-boundary singularities of the form . Trace formulae relating the boundary value v0 of the nonsingular part of the potential to spectral data are derived. Their potential is illustrated by applying them to a number of Schr?dinger problems with singular potentials.

Comment on "Casimir force in the O(n→∞) model with free boundary conditions".

We briefly review recent results of exact calculations of critical Casimir forces of the O(n) ϕ4 model as n→∞ on a three-dimensional strip bounded by two planar free surfaces at a distance L. This model has long-range order below the critical temperature Tc of the bulk phase transition only in the limit L→∞ but remains disordered for all T > 0 for an arbitrary finite strip width L < ∞. A proper description of the system scaling behavior near Tc turns out to be a quite challenging problem because in addition to bulk, boundary, and finite-size critical behaviors, a nontrivial dimensional crossover must be handled. The model admits an exact solution in the limit n → ∞ in terms of the eigenvalues and eigenenergies of a self-consistent Schrödinger equation. This solution contains a potential v(z) with the near-boundary singular behavior v(z → 0+) ≈ −1/(4z2)+ 4m/(π2z), where m = 1/ξ+(|t|) is the inverse bulk correlation length and t ~ (T − Tc)/Tc, and a corresponding singularity at the second boundary plane. In recent joint work with colleagues, the potential v(z), the excess free energy, and the Casimir force were obtained numerically with high precision. We explain how these numerical results can be complemented by exact analytic ones for several quantities (series expansion coefficients of v(z), the scattering data of v(z) in the semi-infinite case L = ∞ for all m >/< 0, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force) by a combination of boundary-operator and short-distance expansions, proper extensions of the inverse scattering theory, new trace formulas, and semiclassical expansions.