The O(n) ${\varphi }^4$ model with free surfaces in the large-$n$ limit: Some exact results for boundary critical behaviour, fluctuation-induced forces and distant-wall corrections

Abstract

The $O(n)$ ${\varphi }^4$ model on a slab ${\mathbb{R}}^{d-1}\times [0,L]$ bounded by free surfaces is studied for $2<d<4$ in the limit $n\to \infty$. The self-consistent potential $V(z)$ which the exact $n\to \infty$ solution of the model involves is analysed by means of boundary operator expansions. Building on the known exact $n\to \infty$ solution for $V(z)$ in the semi-infinite case $L=\infty$ at the bulk critical point, we exactly determine two types of corrections to this potential: (i) those linear in the temperature scaling field $t$ at $L=\infty$, and (ii) the leading $L$-dependent (distant-wall) corrections at the critical point. From (i) exact analytical results at $d=3$ are obtained for the leading temperature singularity of the excess surface free energy and the implied asymptotic behaviours of the scaling functions ${\Theta }_3(x)$ and ${\vartheta }_3(x)$ of the residual free energy $f_{\mathrm{res}}=L^{1-d}{\Theta }_d(tL)$ and the critical Casimir force $\beta {\mathcal{F}}_{\mathrm{C}}(T,L)=L^{-d}{\vartheta }_d(tL)$ in the limit $x\to 0\pm$. The second derivative ${\vartheta }_3^{\prime \prime }(0)$ is computed exactly.

published version, some misprints corrected

References in corpus (5)

• The Casimir effect: from quantum to critical fluctuations
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Large-$n$ approach to thermodynamic Casimir effects in slabs with free surfaces
• Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces
• Fluctuation-induced forces in strongly anisotropic critical systems

Cited by in corpus (10)

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• Fluctuation-induced forces in confined ideal and imperfect Bose gases
• Dispersion Estimates for One-Dimensional Schrödinger Equations with Singular Potentials
• Critical and near critical phase behaviour and interplay between the thermodynamic Casimir and van der Waals forces in confined non-polar fluid medium with competing surface and substrate potentials
• Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• Inverse scattering-theory approach to the exact large-$n$ solutions of $O(n)$ ${\varphi }^4$ models on films and semi-infinite systems bounded by free surfaces
• Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials
• Reply to Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• The three-dimensional $O(n)$ ${\varphi }^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit $n\to \infty$