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Spin correlation function in 2D statistical mechanics models with inhomogeneous line defects

We consider the critical spin-spin correlation function of the 2D Ising model with a line defect which strength is an arbitrary function of position. By using path-integral techniques in the continuum description of this model in terms of fermion fields, we obtain an analytical expression for the correlator as functional of the position dependent coupling. Thus, our result provides one of the few analytical examples that allows to illustrate the transit of a magnetic system from scaling to non-scaling behavior in a critical regime. We also show that the non-scaling behavior obtained for the spin correlator along a non-uniformly altered line of an Ising model remains unchanged in the Ashkin-Teller model.