Fig. 3. Anomalous scaling behaviors of the critical non-Hermitian skin effect.

a Scale-free open-boundary skin eigenstate of the largest $\mathrm{Im}\left[E\right]$ eigenenergy of H_2-chain at system sizes N = 20, 40, 60, and 80 (red, purple, blue, green, respectively). Its rescaled profile, despite decaying exponentially rather than power-law, remains invariant across different N. This scale invariance persists in the N > 20 regime, and is due to the N⁻¹ decay (dashed line) of the inverse skin depth (red dots), as plotted in the inset. Parameters follow Fig. 2’s, except with t₀ = 10⁻³. b Entanglement entropy S (blue) of a half-filled open-boundary H_2-chain at odd system sizes N, with real-space cut at $\lfloor \frac{N}{2}\rfloor$ (the floor function of N/2) and parameters t₁ = 0.58, V = 1, t₀ = 0.4, and δ_a = − δ_b = 0.25. It saturates near zero in the gapless decoupled small N regime, but scales like $~\frac{1}{3}\log N$ (yellow) in the gapless coupled large N regime.