Marginal Extended Perturbations in Two Dimensions and Gap-Exponent Relations

TL;DR

This paper derives the most general form of marginal extended perturbations in two-dimensional systems, calculates first-order corrections to local exponents and energy gaps, and confirms the validity of gap-exponent relations in the perturbed Ising model.

Contribution

It introduces a comprehensive framework for marginal extended perturbations in 2D systems and verifies gap-exponent relations through explicit calculations in the Ising model.

Findings

First-order corrections to local exponents depend on defect amplitude.

First-order corrections to energy gaps are computed for the Ising model.

Gap-exponent relations remain valid under marginal extended perturbations.

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.