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Physical Review Letters

Order-Parameter Profile in Semi-Infinite Systems at Criticality

Joseph Rudnick, David Jasnow

Published Apr 19, 1982• 46 citations
Publisher

Abstract

A description of the decay into the bulk of surface perturbations in a system at its critical point is presented. The description, justified by detailed renormalization-group calculations, yields asymptotic order-parameter decay, ${z}^{\ensuremath{-}\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{2}}$, along with power-law behavior connected with surface exponents under suitable conditions. Comparison with the phenomenological theories of Widom and of Fisher and de Gennes is also made.

DOI
10.1103/physrevlett.48.1059

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A description of the decay into the bulk of surface perturbations in a system at its critical point is presented. The description, justified by detailed renormalization-group calculations, yields asymptotic order-parameter decay, z\ensuremath−(d\ensuremath−2+\ensuremathη)2{z}^{\ensuremath{-}\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{2}}z\ensuremath−2(d\ensuremath−2+\ensuremathη)​, along with power-law behavior connected with surface exponents under suitable conditions. Comparison with the phenomenological theories of Widom and of Fisher and de Gennes is also made.
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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.
The inhomogeneous nature of critical phenomena in semi-infinite systems is studied by calculating the full crossover for both the two-point correlation function and the surface susceptibility ${\ensuremath{\chi}}_{11}$(z) as a function of the distance z from the surface at z=0 for a semi-infinite system near the special transition. We introduce z-dependent counterterms to derive the renormalization-group (RG) equation describing this full crossover. The solution to the RG equation in the one-loop approximation is obtained explicitly and in the long-distance regime is shown to approach bulk behavior exponentially fast. In contrast to the usual approach, our RG equation also contains all information about the short-distance (near-surface) behavior of the theory and its crossover to the bulk. Applications of our method to the descriptions of crossovers for other problems of critical phenomena with several competing lengths are also discussed.
An introduction into the theory of boundary critical phenomena and the application of the field-theoretical renormalization group method to these is given. The emphasis is on a discussion of surface critical behavior at bulk critical points of magnets, binary alloys, and fluids. Yet a multitude of related phenomena are mentioned. The most important distinct surface universality classes that may occur for a given universality class of bulk critical behavior are described, and the respective boundary conditions of the associated field theories are discussed. The short-distance singularities of the order-parameter profile in the diverse asymptotic regimes are surveyed.
A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The d=2d=2d=2 square lattice and the d=3d=3d=3 bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for d=3d=3d=3 are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.