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Physical review. B, Condensed matter

Critical behavior near a symmetry-breaking surface and the stress tensor

Erich Eisenriegler, Michael Stapper

Published Oct 1, 1994• 47 citations
Publisher

Abstract

The critical behavior near a surface that breaks the order parameter symmetry is analyzed with field-theoretic renormalization-group methods. We study the short-distance expansions of the order parameter and the energy density about the surface and investigate various density profiles and correlation functions in the half-space and film geometries. Our results confirm Cardy's conjecture that the stress-energy tensor appears as the leading surface density. Its universal prefactor $C$ in the expansions determines several physical properties of interest in the critical adsorption of fluids: In a film at the bulk critical temperature ${T}_{c}$, $C$ determines the amplitude ratio of the Fisher-de Gennes distant-wall correction to the order-parameter profile and the Casimir correction to the free energy. In the half-space close to ${T}_{c}$, $C$ determines the amplitude ratio of the ${(\frac{z}{\ensuremath{\xi}})}^{d}$ term in the order parameter profile to the Ferer-Stauffer-Wortis term in the bulk free energy. Here $z$ is the distance from the surface and $\ensuremath{\xi}$ the bulk correlation length. In the half-space at ${T}_{c}$, $C$ also determines the universal short-distance amplitudes of layer susceptibilities, which should be observable in computer experiments. We calculate $C$ to second order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and estimate its value for $d=3$. $C$ is closely related to the stress-stress correlation in a half-space, for which we obtain the bulk to surface crossover. We present a simple argument showing why this crossover depends, in general, on the surface universality class except in two dimensions.

DOI
10.1103/physrevb.50.10009

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The critical behavior near a surface that breaks the order parameter symmetry is analyzed with field-theoretic renormalization-group methods. We study the short-distance expansions of the order parameter and the energy density about the surface and investigate various density profiles and correlation functions in the half-space and film geometries. Our results confirm Cardy's conjecture that the stress-energy tensor appears as the leading surface density. Its universal prefactor CCC in the expansions determines several physical properties of interest in the critical adsorption of fluids: In a film at the bulk critical temperature Tc{T}_{c}Tc​, CCC determines the amplitude ratio of the Fisher-de Gennes distant-wall correction to the order-parameter profile and the Casimir correction to the free energy. In the half-space close to Tc{T}_{c}Tc​, CCC determines the amplitude ratio of the (z\ensuremathξ)d{(\frac{z}{\ensuremath{\xi}})}^{d}(\ensuremathξz​)d term in the order parameter profile to the Ferer-Stauffer-Wortis term in the bulk free energy. Here zzz is the distance from the surface and \ensuremathξ\ensuremath{\xi}\ensuremathξ the bulk correlation length. In the half-space at Tc{T}_{c}Tc​, CCC also determines the universal short-distance amplitudes of layer susceptibilities, which should be observable in computer experiments. We calculate CCC to second order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d and estimate its value for d=3d=3d=3. CCC is closely related to the stress-stress correlation in a half-space, for which we obtain the bulk to surface crossover. We present a simple argument showing why this crossover depends, in general, on the surface universality class except in two dimensions.
The critical behavior of the energy density near surfaces belonging to the surface universality classes called ordinary (O) and special (SB) transitions is investigated field theoretically. Both energy density profiles in a film and correlation functions in a half space of the energy density with other operators are considered right at O or SB criticality. We also discuss half-space profiles off criticality. Their properties are in accordance with the operator short-distance expansion of the energy density about the surface. Particular attention is paid to the occurrence of the component perpendicular to the surface of the stress tensor as a surface operator in these expansions.
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.
The surface critical behavior of semi-infinite systems belonging to the Ising universality class with short-range interactions is investigated for supercritical surface enhancement -c>0 and vanishing surface field h1{\mathit{h}}_{1}h1​. Renormalization-group improved perturbation theory is applied to the standard semi-infinite scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ model in d=4-\ensuremath{\epsilon} dimensions to compute the order-parameter profile to one-loop order both for temperatures T with \ensuremath{\tau}\ensuremath{\equiv}(T-Tcb{\mathit{T}}_{\mathit{c}\mathit{b}}Tcb​)/Tcb{\mathit{T}}_{\mathit{c}\mathit{b}}Tcb​\ensuremath{\gtrsim}0 and \ensuremath{\tau}\ensuremath{\lesssim}0. The associated scaling functions are found to cross smoothly over from their short-distance behavior for distances z\ensuremath{\ll}${\ensuremath{\xi}}_{\mathit{b}}$ (=bulk correlation length) to their long-distance behavior for z\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{b}}$ without showing the peculiar nonmonotonic behavior asserted by Peliti and Leibler [J. Phys. C 16, 2635 (1983)]. Furthermore, the short-distance behavior of the profiles is shown to be fully consistent with a \ensuremath{\Vert}\ensuremath{\tau}${\mathrm{\ensuremath{\Vert}}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ singularity of the surface magnetization m1{\mathit{m}}_{1}m1​ plus a regular background term; that is, in contrast to results published recently by other authors, the amplitudes A+{\mathit{A}}_{+}A+​ and ${\mathit{A}}_{\mathrm{\ensuremath{-}}}$ of the contributions ${\mathit{A}}_{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\tau} to m1{\mathit{m}}_{1}m1​ linear in \ensuremath{\tau}>0 or \ensuremath{\tau}0 agree to one-loop order. Finally, we confirm that the universal profiles for the critical adsorption of fluids (governed by the critical-adsorption fixed point at c=+\ensuremath{\infty} and h1{\mathit{h}}_{1}h1​=\ensuremath{\infty}) agree with the previous ones pertaining to the negative-c-transition fixed point at c=-\ensuremath{\infty} and h1{\mathit{h}}_{1}h1​=0.
I use an exact variational formulation of Mikheev and Fisher to study the critical Ising strip with a grain boundary and confining surfaces characterized by arbitrary and different surface magnetic fields. Energy density profiles that serve as order parameters of the system within the used method show strong nonmonotonous behavior in the vicinity of confining surfaces. I consider short-distance expansion of energy density profiles. New universal amplitudes associated with distant-wall corrections exhibit nontrivial crossover behaviors from positive to negative values as the field's variables are continuously varied. Casimir amplitudes calculated in a self-contained manner are characterized by complex manifolds which may comprise two disconnected positive wings separated by an area of negative values. I define and determine the generalized de Gennes-Fisher amplitude, which strongly suggests that a stress tensor is present within one of the distant-wall corrections. This is an unanticipated result given that a similar discovery for standard extraordinary (E) and ordinary (O) surface universality classes was based on the conformal invariance symmetry, which is broken under the present boundary conditions. A grain boundary essentially influences both energy density profiles and Casimir amplitudes, besides surface fields with a number of accompanying features that are examined in detail. We present closed analytic forms of energy density profiles for standard (N) and (O) BCs: ɛNN(x,g),ɛOO(x,g),ɛNO(x,g) [(N) is the normal surface universality class, g is the grain boundary's strength, x is the relative distance x:=z/L with L as a film width], thus enabling us to deduce their important symmetry properties and to have a detailed insight into their general behaviors.
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}.
Two-dimensional Ising strips subject to identical surface fields h(1)=h(2) > or =0 are studied for temperatures above and below the bulk critical temperature T(c) and a range of bulk fields h by means of the density-matrix renormalization-group method. In the case of nonvanishing surface fields, the near-critical behavior of the solvation force f(solv), total adsorption Gamma, inverse longitudinal correlation length xi(parallel)( -1) and specific heat C(H) is strongly influenced by the (pseudo) capillary condensation that occurs below T(c). We obtain scaling functions of f(solv), Gamma, and xi(parallel)(-1). C(H) exhibits a weakly rounded singularity on crossing the pseudocoexistence line. We contrast these results with those for the case of free boundaries where, for temperatures slightly below T(c), f(solv) and C(H) exhibit a sharp extremum away from h=0. Our results have direct repercussions for the properties of near-critical Ising films in three dimensions and we argue that the long-ranged solvation (Casimir) force in confined fluids should be more attractive in the neighborhood of the capillary critical point than exactly at the bulk critical point.