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

Abstract

The $O(n)$ ${\varphi }^4$ 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 $L=\infty$ 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 $x=t(L/{\xi }_{+}{)}^{1/\nu }$ are determined for $x\to -\infty$. In addition, the asymptotic $x\to -\infty$ forms of the universal finite-size scaling functions $\Theta (x)$ and $\vartheta (x)$ of the residual free energy and the Casimir force are computed exactly to order $1/x$, including their $x^{-1}\ln |x|$ anomalies.

arXiv comment53 pages, 4 figures, pdf-LaTeX; v2: eq. (7.73) is simplified; v3: misprints corrected

References in corpus (28)

• Critical Phenomena and Renormalization-Group Theory
• The `Friction' of Vacuum, and other Fluctuation-Induced Forces
• Quantum Field Theory in the Large N Limit: a review
• The Theory of Boundary Critical Phenomena
• Conformal Field Theories Near a Boundary in General Dimensions
• The Casimir effect: from quantum to critical fluctuations
• Massive Field-Theory Approach to Surface Critical Behavior in Three-Dimensional Systems
• Ordinary, extraordinary, and normal surface transitions: extraordinary-normal equivalence and simple explanation of $|T-T_c{|}^{2-\alpha }$ singularities
• Lattice Green functions in all dimensions
• Crossover from Attractive to Repulsive Casimir Forces and Vice Versa
• Critical Casimir Effect in superfluid wetting films
• Thinning of superfluid films below the critical point
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Aspect-ratio dependence of thermodynamic Casimir forces
• Surface critical behavior in fixed dimensions $d<4$: Nonanalyticity of critical surface enhancement and massive field theory approach
• Excess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results
• Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory
• Exact Three Dimensional Casimir Force Amplitude, $C$-function and Binder's Cumulant Ratio: Spherical Model Results
• 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
• Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions
• Casimir force in the O(n -> infinity) model with free boundary conditions
• Critical Casimir amplitudes for $n$-component ${\varphi }^4$ models with O(n)-symmetry breaking quadratic boundary terms
• Pronounced minimum of the thermodynamic Casimir forces of O($\mathbf{n}$) symmetric film systems: Analytic theory
• 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
• Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials

Cited by in corpus (5)

• Critical Casimir Effect: Exact Results
• Fluctuation-induced forces in confined ideal and imperfect Bose gases
• Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials
• 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 $n\to \infty$
• A formula for eigenvalues of Jacobi matrices with a reflection symmetry