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Marginalextended perturbations in two dimensions and gap–exponent relations

Loïc Turban, Bertrand Berche

Published Jan 1, 1993• 6 citations
OpenAlex

Abstract

Abstract. The most general form of a marginal extended perturbation in a two– dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for

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Marginal extended perturbations in two dimensions and gap-exponent relations

Loïc Turban, Bertrand Berche

The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.

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Radial Fredholm perturbation in the two-dimensional Ising model and gap-exponent relation

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We consider concentric circular defects in the two-dimensional Ising model, which are distributed according to a generalized Fredholm sequence, i.e. at exponentially increasing radii. This type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation. The critical exponent of the local magnetization is obtained through finite-size scaling, using a corner transfer matrix approach in the extreme anisotropic limit. It varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial Hilhorst-van Leeuwen model. Through a conformal mapping of the system onto a strip, the gap-exponent relation is shown to remain valid for such an aperiodic defect.

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The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.
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 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.
We consider concentric circular defects in the two-dimensional Ising model, which are distributed according to a generalized Fredholm sequence, i.e. at exponentially increasing radii. This type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation. The critical exponent of the local magnetization is obtained through finite-size scaling, using a corner transfer matrix approach in the extreme anisotropic limit. It varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial Hilhorst-van Leeuwen model. Through a conformal mapping of the system onto a strip, the gap-exponent relation is shown to remain valid for such an aperiodic defect.