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Physical Review Letters

Absence of hyperuniversality in critical films

Erich Eisenriegler, M. Krech, S. Dietrich

Published Feb 1, 1993• 36 citations
Publisher

Abstract

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 ${\mathit{T}}_{\mathit{c},}$b in spatial dimension d=4-\ensuremath{\varepsilon}. At distances z\ensuremath{\ll}L from one wall a the profile contains a perturbation ${\mathit{B}}_{\mathit{a},}$b(z/L${)}^{\mathit{d}}$ induced by the distant wall b. In contrast to previous expectations the ratio of the universal amplitude ${\mathit{B}}_{\mathit{a},}$b and the corresponding Casimir amplitude is not hyperuniversal, i.e., not independent of both surface universality classes characterizing the walls a and b.

DOI
10.1103/physrevlett.70.619

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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.
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}.
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.
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.
Two-dimensional Ising films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature Tc{T}_{c}Tc​ for two choices of surface fields (a) H1=HL=0{H}_{1}{=H}_{L}=0H1​=HL​=0 (ordinary transition), and (b) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ and for the analog of the solvation force fsolv{f}_{\mathrm{solv}}fsolv​ are found and discussed. When H1=0{H}_{1}=0H1​=0 the scaling function for fsolv{f}_{\mathrm{solv}}fsolv​ has two symmetric minima at $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ with an amplitude at the minimum about 3.8 times the value at H=0,H=0,H=0, the Casimir amplitude. For the normal transition the scaling function for fsolv{f}_{\mathrm{solv}}fsolv​ has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.
We apply generalizations of two-scale-factor universality, or hyperuniversality, to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in two- and three-dimensional Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. We also give a general proof for the existence of universal jumps in two-dimensional superfluids at finite temperature using finite-size hyperuniversality.