Anomalous Skin Effects in Disordered Systems with a Single non-Hermitian Impurity

We explore anomalous skin effects at non-Hermitian impurities by studying their interplay with potential disorder and by exactly solving a minimal lattice model. A striking feature of the solvable single-impurity model is that the presence of anisotropic hopping terms can induce a scale-invariant accumulation of all eigenstates opposite to the bulk hopping direction, although the non-monotonic behavior is fine-tuned and further increasing such hopping weakens and eventually reverses the effect. The interplay with bulk potential disorder, however, qualitatively enriches this phenomenology leading to a robust non-monotonic localization behavior as directional hopping strengths are tuned. These phenomena persist even in the limit of an entirely Hermitian bulk with a single non-Hermitian impurity.