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

Hybrid scale-free skin effect in non-Hermitian systems: A transfer matrix approach