Study of line defects in infinite networks of resistors

We study infinite resistor networks perturbed by line defects, in which the resistances are periodically modified along a single line. Using the Sherman–Morrison identity applied to the lattice Laplace operator, we develop a general analytical framework for computing the Green's function and the equivalent resistance between arbitrary nodes. The resulting expression is a one-dimensional integral that is evaluated exactly in special cases. While our analysis is carried out for the square lattice, the method readily extends to other lattice geometries and networks with general impedances. Therefore, this framework is useful for studying the boundary behavior of topolectrical circuits, which serve as classical analogs of topological insulators.