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Physical Review Letters

Universal critical-point amplitudes in parallel-plate geometries

John Cardy

Published Sep 17, 1990• 89 citations
Publisher

Abstract

In a critical system contained between parallel plates, it is known that the magnetization and energy density profiles close to one wall are modified due to the presence of the other wall. In addition, there is a long-range force between the walls. It is shown that the ratios of the ampiltudes governing these effects are universal, and moreover, independent of the boundary conditions at either wall. Their values are proportional to the scaling dimension of the density under consideration. Results are also given for the effects of local curvature of the wall on the density profiles. These results are shown to be a consequence of the conformal invariance of the critical system.

DOI
10.1103/physrevlett.65.1443

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In a critical system contained between parallel plates, it is known that the magnetization and energy density profiles close to one wall are modified due to the presence of the other wall. In addition, there is a long-range force between the walls. It is shown that the ratios of the ampiltudes governing these effects are universal, and moreover, independent of the boundary conditions at either wall. Their values are proportional to the scaling dimension of the density under consideration. Results are also given for the effects of local curvature of the wall on the density profiles. These results are shown to be a consequence of the conformal invariance of the critical system.
Scaling density profiles O(N) symmetric systems confined between two parallel plates with a large separation L are analyzed by field-theoretical methods for a continuous phase transition at bulk criticality Tc,{\mathit{T}}_{\mathit{c},}Tc,​b in spatial dimension d=4-\ensuremath{\varepsilon}. At distances z\ensuremath{\ll}L from one wall a the profile contains a perturbation Ba,{\mathit{B}}_{\mathit{a},}Ba,​b(z/L)d{)}^{\mathit{d}})d induced by the distant wall b. In contrast to previous expectations the ratio of the universal amplitude Ba,{\mathit{B}}_{\mathit{a},}Ba,​b and the corresponding Casimir amplitude is not hyperuniversal, i.e., not independent of both surface universality classes characterizing the walls a and b.
The conformal-invariance approach is extended to two-dimensional critical systems with mixed ab boundary conditions. The profiles 〈\ensuremath{\varphi}(r)〉ab{\mathrm{〉}}_{\mathit{a}\mathit{b}}〉ab​ of the magnetization and the energy density in Ising and Q-state Potts strips with +- and fixed-free boundary conditions are derived. The profile of an arbritrary operator degenerate at level 2 is expressed in terms of hypergeometric functions. The universal amplitudes Aab{\mathit{A}}_{\mathit{a}\mathit{b}}Aab​Bab{\mathit{B}}_{\mathit{a}\mathit{b}}Bab​ of the Casimir force and the Fisher--de Gennes distant-wall correction to the profile satisfy Bab{\mathit{B}}_{\mathit{a}\mathit{b}}Bab​=-4\ensuremath{\pi}${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$Aab{\mathit{A}}_{\mathit{a}\mathit{b}}Aab​. Here ${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$ is the scaling dimension of \ensuremath{\varphi}(r) and c is the central charge.
Scaling density profiles of critical O(N)-symmetric systems confined to a film geometry are analyzed by field-theoretic renormalization-group techniques in d=4-\ensuremath{\varepsilon} spatial dimensions. At bulk criticality we explicitly calculate the energy density profile for five different boundary conditions and determine their corresponding universal scaling functions. In order to discuss the influence of the boundary conditions at the two surfaces of the film on the global shape of the profiles, a suitable exponentiation scheme is constructed. Using these exponentiated scaling functions as well as results from conformal field theory, the shapes of the energy density profiles in d=2,3, and 4 are compared. Finally, we discuss the short-distance behavior of the energy density profiles close to the surfaces of the film as it follows from the knowledge of the full profiles in first order \ensuremath{\varepsilon}.
A critical phase confined between two parallel plates, with symmetry-breaking fields h1{h}_{1}h1​ and h2{h}_{2}h2​ acting on each plate, is considered for (i) h1h2>0{h}_{1}{h}_{2}>0h1​h2​>0 (denoted ab=++ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}++ab=++) and (ii) h1h2<0{h}_{1}{h}_{2}<0h1​h2​<0 ( $ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{-}$). Using local-functional methods, we calculate order-parameter scaling functions ${\ensuremath{\Psi}}_{\mathrm{ab}}(x)$ and Casimir amplitudes Aab{A}_{\mathrm{ab}}Aab​, for general dimension ddd. At d=2d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2d=2, our ${\ensuremath{\Psi}}_{++}(x)$ almost coincides with exact conformal predictions. For the Ising universality class, we obtain expansions for Aab{A}_{\mathrm{ab}}Aab​ in $\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d\ensuremath{\downarrow}0$ in excellent agreement with those obtained from field theory and, in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3, new results are presented for Aab{A}_{\mathrm{ab}}Aab​ and ${\ensuremath{\Psi}}_{\mathrm{ab}}(x)$.
A critical Ising strip with confining boundaries characterized by arbitrary and different surface magnetic field variables is considered by the exact variational formulation of Mikheev and Fisher. As one of the most important scaling densities in films we study the universal functions of the energy density profiles which exhibit strong nonmonotonous behavior near the confining boundaries. We also examine their short-distance expansion that reveals novel universal amplitudes associated with the distant-wall corrections. They manifest nontrivial crossover behaviors from positive to negative values as the fields variables are freely varied. As an indispensable part of our new formulation of the generalized distant-wall correction de Gennes-Fisher universal amplitude we derive in a self-contained manner within the present approach the Casimir amplitudes that may change their nature twice under the strong influence of surface fields: from repulsive to attractive and vice versa along certain trajectories. Our findings strongly suggest a surprising role of the stress-tensor within one of the distant-wall corrections, given that similar discovery for standard extraordinary and ordinary surface universality classes was based on the conformal invariance symmetry which is broken under the present boundary conditions. We also show that a constituent part of the de Gennes-Fisher amplitude, defined in a generalized sense, is not hyperuniversal in two dimensions, contrary to earlier results for standard boundary conditions in this dimension.