  Surpassing the individual characteristics of the non-Hermitian skin effect
(NHSE) and the scale-free (SF) effect observed recently, we systematically
exploit the exponential decay behavior of bulk eigenstates via the transfer
matrix approach in non-Hermitian systems. We concentrate on one-dimensional
(1D) finite-size non-Hermitian systems with 2*2 transfer matrices in either the
absence or presence of the boundary impurity. We analytically unveil that the
unidirectional SF effect emerges with the singular transfer matrices, while the
hybrid scale-free skin (SFS) effect appears with the nonsingular transfer
matrices even when an open boundary condition (OBC) is imposed. The
unidirectional SF effect exceeds the scope of the SF effect in previous works,
while the hybrid SFS effect is an interesting interplay between the skin effect
and the SF effect in finite-size systems. Our results reveal that the skin
effect under the OBC prevails when it coexists with the SF effect as the system
approaches the thermodynamic limit in the presence of the hybrid SFS effect.
Our approach paves the way for rigorous and unified explorations of the skin
and SF effects in both Hermitian and non-Hermitian systems with generic
boundary conditions.
