Critical behaviour of semi-infinite systems

Bray, A. J.; Moore, M. A.

doi:10.1088/0305-4470/10/11/021

Cited by 308 publications

(265 citation statements)

References 44 publications

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“…Otherwise one can try recovering the equivalence between the spherical model and O(n) spin models by imposing spherical constraints ensuring the same mean square value for all spins of the system (Knops 1973). In the case of a film geometry this is equivalent to having a spherical constraint on each layer of the system with a space dependent spherical field along the finite direction (Hikami & Abe 1976, Bray & Moore 1977, Ohno & Okabe 1983) whose relaxed version reduces to the model under consideration. Even in this case an accurate study of the problems related to finite-size scaling remains rather untractable analytically (see e.g.…”

Section: Discussionmentioning

confidence: 99%

Finite-size effects in the spherical model of finite thickness

Chamati¹

2008

J. Phys. A: Math. Theor.

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Abstract.A detailed analysis of the finite-size effects on the bulk critical behaviour of the d-dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N ) and a combination of Neumann and Dirichlet (N D) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for (p) and (a).

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Section: Discussionmentioning

confidence: 99%

Finite-size effects in the spherical model of finite thickness

Chamati¹

2008

J. Phys. A: Math. Theor.

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“…Of particular interest is the (d = 3)-dimensional case on which we now focus. For it, a number of exact analytic results have been obtained for the classical theory with DDBCs [26,27,[30][31][32][33][34][35], which are known to apply to this theory with free BCs at d = 3 asymptotically in the large length scale limit. Using a combination of techniques such as direct solutions of the self-consistent equations [32], short-distance and boundary-operator expansions [27], trace formulas [34], inverse scattering methods for the semi-infinite case D = ∞ and matched semiclassical expansions for D < ∞ [33], exact analytic results for several series expansion coefficients of the self-consistent potential v * (z) and for the asymptotic x → ∞ behaviors of the eigenvalues ε DD ν , eigenfunctions h DD ν , and the classical scaling functions of the residual free energy and the Casimir force have been determined.…”

Section: Generalizing the Imperfect Bose Gas Model To Allow For Nomentioning

confidence: 99%

“…IV D. Its scaling function Θ DD ∞,3 (tD) may be obtained for all values of the scaling field t ∝ µ c − µ 0 from the numerical solution of the O(∞) φ 4 model determined in [26][27][28][29]. Furthermore, a variety of exact analytical results may be inferred from those known for the O(∞) φ 4 model subject to DDBCs [30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Fluctuation-induced forces in confined ideal and imperfect Bose gases

Diehl

Rutkevich

2017

Phys. Rev. E

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Fluctuation-induced ("Casimir") forces caused by thermal and quantum fluctuations are investigated for ideal and imperfect Bose gases confined to d-dimensional films of size ∞ d−1 × D under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions, are determined for the ideal gas case with these BCs, where λ th and ξ are the thermal de-Broglie wavelength and the bulk correlation length, respectively. The associated limiting scaling functionsξ ) describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free O(2) theory for BC = P, A, DD, DN, NN. For d = 3, they are expressed in closed analytical form in terms of polylogarithms. The analogous scaling functions Υ BC d (x λ , x ξ , c1D, c2D) and Θ R d (x ξ , c1D, c2D) under the RBCs (∂z − c1)φ|z=0 = (∂z + c2)φ|z=D = 0 with c1 ≥ 0 and c2 ≥ 0 are also determined. The corresponding scaling functions Υ P ∞,d (x λ , x ξ ) and Θ P ∞,d (x ξ ) for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with n internal degrees of freedom in the limit n → ∞. Hence, for d = 3, Θ P ∞,d (x ξ ) is known exactly in closed analytic form. To account for the breakdown of translation invariance in the direction perpendicular to the boundary planes implied by free BCs such as DDBCs, a modified imperfect Bose gas model is introduced that corresponds to the limit n → ∞ of this interacting Bose gas. Numerically and analytically exact results for the scaling function Θ DD ∞,3 (x ξ ) therefore follow from those of the O(2n) φ 4 model for n → ∞.

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“…In the definition of the excess adsorption [Eq. (10)] one considers a finite volume V of integration that is enlarged to fill the total volume of the wedge or ridge in the thermodynamic limit. As shown below for two examples, the expression for the line term (such as Eq.…”

Section: Appendix: Decomposition Of the Excess Adsorptionmentioning

confidence: 99%

“…The transition to this state from the disordered state is usually denoted as the extraordinary transition. Bray and Moore [10] predicted an equivalence between the normal and the extraordinary transitions that was later proved by Burkhardt and Diehl [11]. In contrast to the ordinary transition in magnetic systems, the corresponding extraordinary transition has been investigated to a lesser extent.…”

Section: Introductionmentioning

confidence: 96%

Critical adsorption and Casimir torque in wedges and at ridges

Palágyi

Dietrich

2004

Phys. Rev. E

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Geometrical structures of confining surfaces profoundly influence the adsorption of fluids upon approaching a critical point Tc in their bulk phase diagram, i.e., for t = (T −Tc)/Tc → ±0. Guided by general scaling considerations, we calculate, within mean-field theory, the temperature dependence of the order parameter profile in a wedge with opening angle γ < π and close to a ridge (γ > π) for T ≷ Tc and in the presence of surface fields. For a suitably defined reduced excess adsorption Γ±(γ, t → ±0) ∼ Γ±(γ)|t| β−2ν we compute the universal amplitudes Γ±(γ), which diverge as Γ±(γ → 0) ∼ 1/γ for small opening angles, vary linearly close to γ = π for γ < π, and increase exponentially for γ → 2π. There is evidence that, within mean-field theory, the ratio Γ+(γ)/Γ−(γ) is independent of γ. We also discuss the critical Casimir torque acting on the sides of the wedge as a function of the opening angle and temperature.

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Critical behaviour of semi-infinite systems

Cited by 308 publications

References 44 publications

Finite-size effects in the spherical model of finite thickness

Finite-size effects in the spherical model of finite thickness

Fluctuation-induced forces in confined ideal and imperfect Bose gases

Critical adsorption and Casimir torque in wedges and at ridges

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