  This work develops a full framework for non-Hermitian impurity physics in a
non-reciprocal lattice, with PBCs, OBCs and even their interpolations being
special cases across a whole range of boundary impurity strengths. As the
impurity strength is tuned, the localization of steady states can assume very
rich behavior, including the expected non-Hermitian skin effect, Bloch-like
states albeit broken translational invariance, and surprisingly, scale-free
accumulation along or even against the direction of non-reciprocity. We further
uncover the possibility of the co-existence of non-Hermitian skin effect and
scale-free localization, where qualitative aspects of the system's spectrum can
be extremely sensitive to impurity strength. We have also proposed specific
circuit setups for experimental detection of the scale-free accumulation, with
simulation results confirming our main findings.

Mesoscale and Nanoscale Physics;Quantum Physics