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arXiv:hep-th/9305119 · Journal of Physics A Mathematical and General

Marginal extended perturbations in two dimensions and gap-exponent relations

Loïc Turban, Bertrand Berche

Published Jul 7, 1993• 9 citations
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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 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.

DOI
10.1088/0305-4470/26/13/017

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The conformal mapping w=(L/2\ensuremath{\pi})lnz transforms the critical plane with a radial perturbation \ensuremath{\alpha}${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}\mathit{y}}$ into a cylinder with width L and a constant deviation \ensuremath{\alpha}(2\ensuremath{\pi}/L)y{)}^{\mathit{y}})y from the bulk critical point when the decay exponent y is such that the perturbation is marginal. From the known behavior of the homogeneous off-critical system on the cylinder, one may deduce the correlation functions and defect exponents on the perturbed plane. The results are supported by an exact solution for the Gaussian model.
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.
We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power κ of the distance l from the free surface with an oscillating amplitude Al=(−1)flAA_l = ( - 1)^{f_l } AAl​=(−1)fl​A where f l = 0, 1 follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent κ, the wandering exponent ω which governs the fluctuation of the sequence and the bulk correlation length exponent ν. Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the (κ, ω)-plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed.
We study the critical energy and magnetization profiles for the Ising quantum chain with a marginal extended surface perturbation of the form A / y , y being the distance from the surface (Hilhorst-van Leeuwen model). For weak local couplings, A < A c , the model displays a continuous surface transition with A -dependent exponents, whereas, for A > A c , there is surface order at the bulk critical point. If conformal invariance is assumed to hold with such marginal perturbations, it predicts conformal profiles with the same scaling form as for the unperturbed quantum chain, with marginal surface exponents replacing the unperturbed ones. The results of direct analytical and numerical calculations of the profiles confirm the validity of the conformal expressions in the regimes of second- and first-order surface transitions.
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