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Cambridge University Press eBooks

Surface critical behaviour

John Cardy

Published Apr 26, 1996• 1 citation

Abstract

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.

DOI: 10.1017/cbo9781316036440.008

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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 critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.

Abstract ‘The critical point’ in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close to the critical point. The successful theory developed by Ken Wilson exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics involved can be made tractable by considering the abstract idea of a system with a variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.

Systems with a bulk first-order transition can display diverging correlation lengths close to a surface. This surface induced disordering yields a special type of surface criticality. Using extensive numerical simulations we study surface quantities in the two-dimensional Potts model with a large number of states

q

which undergoes a discontinuous bulk transition. The surface critical exponents are thereby found to depend on the value of

q

, which is in contrast to prior claims that these exponents should be universal and independent of

q

. It follows that surface-induced disordering at first-order transitions is characterized by exponents that depend on the details of the model.

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

{\mathit{d}}^{\mathrm{*}}

\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.

This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.