Marginal extended perturbations in two dimensions and gap – exponent relations

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.