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Physical review. B, Condensed matter

Critical phenomena near free surfaces and defect planes

Theodore W. Burkhardt, Erich Eisenriegler

Published Aug 1, 1981• 59 citations
Publisher

Abstract

We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.

DOI
10.1103/physrevb.24.1236

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We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.
The existence of nonuniversal, as well as surface (universal) critical behavior near a defect plane is demonstrated on a one-dimensional quantum lattice gas model, belonging to the two-dimensional Ising universality class. According to numerical results the weak defect region is attracted by a line of fixed points starting at the bulk critical point, while the strong defect region is controlled by a separate fixed point describing the surface transition of the model. This new kind of phase diagram is argued to be general for two-dimensional lattice gas models.
The local critical behavior of the ddd-dimensional nnn-vector model near an internal $d\ensuremath{-}1$ dimensional plane of weakened interactions in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ changes qualitatively at d=3d=3d=3. For d>3d>3d>3 the critical exponent ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$, which characterizes the correlation of spins in the defect plane, is universal, i.e., is independent of the strength of the interactions in the defect plane. We argue that ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ is nonuniversal for d=3d=3d=3 and calculate its dependence on the difference between the defect and bulk couplings to second order in perturbation theory. We show that the excess energy density at a distance zzz from the defect plane decays at the critical point to the bulk value as ${z}^{\ensuremath{-}2}$ for $d\ensuremath{\ge}3$ and faster than ${z}^{\ensuremath{-}2}$ for d<3d<3d<3. Near the free surface of a semi-infinite system one finds a ${z}^{\ensuremath{-}2}$ dependence for d>3d>3d>3 as well as for $d\ensuremath{\ge}3$.
We study the local critical behavior near the free surface of a d-dimensional system with a smooth inhomogeneity induced by a defect of dimension d∗{\mathit{d}}^{\mathrm{*}}d∗\ensuremath{\le}d-1 located on the surface. The couplings deviate from their bulk critical value by \ensuremath{\lambda}${\mathit{r}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ where ${\mathit{r}}_{\mathrm{\ensuremath{\perp}}}$ is the distance to the center of the defect. Using a phenomenological approach, the change in the surface critical exponents is calculated to first order in the perturbation amplitude \ensuremath{\lambda} when the system is marginal, i.e., when the decay exponent of the perturbation \ensuremath{\alpha} is equal to the scaling dimension ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\varepsilon}}}^{0}$ of the bulk energy-density operator. Finally, the critical behavior near extended defects in the bulk is compared to the surface case.
The effects of irrelevant surface perturbations on the local critical behavior of ($4\ensuremath{-}\ensuremath{\epsilon}$)-dimensional nnn-vector models with a free surface or an internal defect plane are investigated with particular attention paid to the corrections to scaling induced by the surface term $\frac{1}{2}c\ensuremath{\int}{\mathrm{surface}}^{}{\ensuremath{\varphi}}^{2}$ near the ordinary transition. A systematic expansion in ${c}^{\ensuremath{-}1}$ is developed. In the case of a semi-infinite system, ${c}^{\ensuremath{-}1}$ couples to the square of the normal derivative of $\ensuremath{\varphi}$. The associated crossover exponent is ${\ensuremath{\Phi}}^{o}=\ensuremath{-}\ensuremath{\nu}$ to all orders in $\ensuremath{\epsilon}$. In the case of an infinite-space model with a defect plane, ${c}^{\ensuremath{-}1}$ couples to the squares of the normal derivatives on either side of the plane and to the product of the normal derivative on one side with the normal derivative on the other side. The latter operator represents perpendicular interactions across the defect plane, while the other two describe parallel interactions between spins on the same side of the plane. The associated crossover exponents are ${\ensuremath{\Phi}}_{\ensuremath{\parallel}}^{o}=\ensuremath{-}\ensuremath{\nu}$ and ${\ensuremath{\Phi}}_{\ensuremath{\perp}}^{o}={\ensuremath{\gamma}}_{11}^{o}$. A reanalysis of Bray and Moore's derivation of the (incorrect) relation ${\ensuremath{\Phi}}^{o}={\ensuremath{\gamma}}_{11}^{o}=\ensuremath{\nu}\ensuremath{-}1$ is presented which reveals where their arguments go wrong.
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.
We consider the critical system with a point defect and study the variation of thermodynamic quantities, which are the differences between those with and without the defect. Within renormalization group theory, we show generally that the critical exponent of the internal energy variation is the specific heat exponent of a pure system, and the critical exponent of the heat capacity variation is that for the temperature derivative of specific heat of a pure system. This conclusion is valid for the isotropic systems with a short-range interaction. As an example we solve the two-dimensional Ising model with a point defect numerically. The variations of the free energy, internal energy, and specific heat are calculated with the bond propagation algorithm. At the critical point, the internal energy variation diverges with the lattice size logarithmically and the heat capacity variation diverges with size linearly. Near the critical point, the internal energy variation behaves as ln|t| and the heat capacity variation behaves as |t|^{-1}, where t is the reduced temperature.