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 TMF, 2017, Volume 190, Number 2, Pages 325–343 (Mi tmf9120)

The three-dimensional $O(n)$ $\phi^4$ model on a strip with free boundary conditions: Exact results for a nontrivial dimensional crossover in the limit $n\to\infty$

H. W. Diehl, S. B. Rutkevich

Department of Physics, University of Duisburg-Essen, Duisburg, Germany

Abstract: We briefly review recent results of exact calculations of critical Casimir forces of the $O(n)$ $\phi^4$ model as $n\to\infty$ 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 $T_{\mathrm c}$ of the bulk phase transition only in the limit $L\to\infty$ but remains disordered for all $T>0$ for an arbitrary finite strip width $L<\infty$. A proper description of the system scaling behavior near $T_{\mathrm c}$ 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 $n\to\infty$ 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 $v(z\to0+)\approx-1/(4z^2)+4m/(\pi^2z)$, where $m=1/\xi_+(|t|)$ is the inverse bulk correlation length and $t\sim(T-T_{\mathrm c})/T_{\mathrm c}$, 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 $L=\infty$ for all $m\gtreqless 0$, 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.

Keywords: fluctuation-induced force, Casimir effect, inverse scattering problem, dimensional crossover, finite-size scaling

DOI: https://doi.org/10.4213/tmf9120

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English version:
Theoretical and Mathematical Physics, 2017, 190:2, 279–294

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Document Type: Article

Citation: H. W. Diehl, S. B. Rutkevich, “The three-dimensional $O(n)$ $\phi^4$ model on a strip with free boundary conditions: Exact results for a nontrivial dimensional crossover in the limit $n\to\infty$”, TMF, 190:2 (2017), 325–343; Theoret. and Math. Phys., 190:2 (2017), 279–294

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf9120
• https://doi.org/10.4213/tmf9120
• http://mi.mathnet.ru/eng/tmf/v190/i2/p325

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This publication is cited in the following articles:
1. Diehl H.W., Rutkevich S.B., “Fluctuation-Induced Forces in Confined Ideal and Imperfect Bose Gases”, Phys. Rev. E, 95:6 (2017), 062112
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