Scale free localization: The role of semi-edges

Abstract In a one-dimensional lattice, a semi-edge refers to a boundary region where one direction of coupling between lattice sites is absent, while the coupling in the opposite direction remains intact. This creates a transitional regime where the lattice does not exhibit a complete edge with zero couplings in both directions, but rather shows partial edge-like characteristics due to the absence of coupling in one direction. Physically, this can manifest as a region where certain properties or behaviors associated with edges, such as localized modes begin to appear due to the asymmetry in coupling. We consider a one-dimensional non-Hermitian lattice and explore skin localization in the presence of a coupling impurity that acts as a semi-edge. Our findings reveal the emergence of scale-free localized (SFL) modes when an impurity significantly reduces coupling in one direction.