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Journal of Physics A Mathematical and General

Non-universal and anomalous surface critical behaviour in an inhomogeneous semi-infinite Gaussian model

Theodore W. Burkhardt, Ihnsouk Guim

Published Jun 1, 1982• 12 citations
Publisher

Abstract

Considers a semi-infinite Gaussian model with spatially inhomogeneous short-range couplings that depend on the distance z from the surface. Far from the surface the coupling constants vary as K(z)=K B -Az -y . For y=>2 the pair correlation function of the surface spins decays as a power law with a universal exponent eta /sub ///=2 at the bulk critical temperature. For y=2, eta /sub /// is non-universal, and for y<2 there is an anomalous exponential decay.

DOI
10.1088/0305-4470/15/6/010

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Considers a semi-infinite Gaussian model with spatially inhomogeneous short-range couplings that depend on the distance z from the surface. Far from the surface the coupling constants vary as K(z)=K B -Az -y . For y=>2 the pair correlation function of the surface spins decays as a power law with a universal exponent eta /sub ///=2 at the bulk critical temperature. For y=2, eta /sub /// is non-universal, and for y<2 there is an anomalous exponential decay.
A semi-infinite nearest-neighbour square Ising lattice is investigated whose couplings J(l) at a distance l from the boundary differ from homogeneity by an amount j(l)-J(infinity) approximately -A/l. On the basis of the Pfaffian method the critical behaviour at the surface of this system is investigated. The exponents eta 11 , nu 11 , beta 1 , gamma 11 and delta 11 all display rich non-universal behaviour as a function of the amplitude A. For A below a critical value there is a spontaneous surface magnetisation when the bulk (l = infinity) is critical and an asymmetry between the exponents on either side of the critical point.
We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large mmm, mmm being the distance from the edge. The case A<0A<0A<0 of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when A>0A>0A>0. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.
Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am -y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v -1 , a surface magnetic phase at the bulk critical temperature for y<v -1 , A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v -1 .
One of the most striking aspects of critical behaviour is that of the crucial role played by the geometry of the System. Critical exponents depend in a non-trivial manner on the dimensionality d . The very existence of a phase transition depends on the way in which the infinite volume limit is taken, as discussed in Section 4.4. This happens because the critical fluctuations, which determine the universal properties, occur at long wavelengths and are therefore very sensitive to the large scale geometry. By contrast, the non-critical properties of a system are sensitive to fluctuations on the scale of the correlation length and are therefore much less influenced. This line of reasoning also suggests that not all points in a system are equivalent in the way the local degrees of freedom couple to these critical fluctuations. So far, we have considered the behaviour of scaling operators only at points deep inside the bulk of a system. Near a boundary, however, the local environment of a given degree of freedom is different, and we might expect to find different critical properties there. In general, such differences should extend into the bulk only over distances of the order of the bulk correlation length. However, at a continuous bulk phase transition, this distance diverges, and we should expect the influence of boundaries to be more pronounced. The simplest modification of the bulk geometry to consider is that of a ( d − 1)-dimensional hyperplane bounding a semi-infinite d -dimensional system.
The addition of a spatial dependence of the form $\ensuremath{\Delta}J={\mathrm{Az}}^{\ensuremath{-}y}$ to the short-range couplings of a semi-infinite spin system with surface at z=0z=0z=0 and bulk exponent $\ensuremath{\nu}$ is predicted to modify the surface critical behavior for $y<{\ensuremath{\nu}}^{\ensuremath{-}1}$ but not for $y>{\ensuremath{\nu}}^{\ensuremath{-}1}$. This information is implied by the scaling relation ${\ensuremath{\varphi}}_{y}=1\ensuremath{-}\ensuremath{\nu}y$ proposed for the crossover exponent. Crossover effects associated with long-range couplings of the surface spins are also discussed.
The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al -2 , where l is the distance from the surface. The surface critical exponents of this system are non-universal. Under a conformal mapping onto a strip of width L, the Al -1 inhomogeneity transforms into A((L/ pi )sin( pi L/L)) -1 . For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit. The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance.