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arXiv:cond-mat/0106602 · Journal of Physics A Mathematical and General

Critical behaviour in parabolic geometries

I Peschel, Loïc Turban, Ferenc Iglói

Published Oct 21, 1991• 24 citations
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Abstract

The authors study two-dimensional systems with boundary curves described by power laws. Using conformal mappings they obtain the correlations at the bulk critical point. Three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions. For an Ising system of parabolic shape the behaviour of the order at the tip is also found.

DOI
10.1088/0305-4470/24/20/004

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The authors study two-dimensional systems with boundary curves described by power laws. Using conformal mappings they obtain the correlations at the bulk critical point. Three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions. For an Ising system of parabolic shape the behaviour of the order at the tip is also found.
The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first-order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.
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 authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T c mod 2- alpha , as the bulk free energy. At bulk criticality energy-energy correlations decay as r -2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.
Using previous results from boundary conformal field theory and integrability, a phase diagram is derived for the two-dimensional Ising model at its bulk tri-critical point as a function of boundary magnetic field and boundary spin-coupling constant. A boundary tri-critical point separates phases where the spins on the boundary are ordered or disordered. In the latter range of coupling constant, there is a non-zero critical field where the magnetization is singular. In the former range, as the temperature is lowered, the boundary undergoes a first-order transition while the bulk simultaneously undergoes a second-order transition.