Energy-resolved coexistence of skin and bulk-localised states in a quasiperiodic non-Hermitian Su–Schrieffer–Heeger chain

We study a Su–Schrieffer–Heeger (SSH) chain with nonreciprocal intracell hopping v ± δ, reciprocal intercell hopping w and a real Aubry–André–Harper (AAH) potential of amplitude λ on both sublattices. For 0 < δ < v an exact open-chain similarity transformation maps the model onto a Hermitian SSH–AAH chain with intracell hopping v_eff = √(v² − δ²), and the right eigenvectors acquire a gauge factor exp(−κj) per unit cell, with κ = ½ ln[(v + δ)/(v − δ)]. An exponentially localised parent bulk state keeps a bulk centre if its Lyapunov exponent γ_H(E) exceeds κ and becomes a skin mode if γ_H(E) < κ. A Herman-type bound, γ_H(E) ≥ ln[λ²/(4 v_eff w)], implies γ_H(E) > κ at all energies for λ > λ_suff = 2√(w(v + δ)). In chains of up to 1600 sites, the measured bulk fraction of right eigenstates approaches this prediction except near the interval edges. The finite-size results support an interval λ₁(δ) < λ < λ₂(δ) of coexisting skin and bulk-localised states that persists towards the thermodynamic limit; within it the bulk fraction grows in steps as subbands cross the threshold, with participation-ratio minima at the crossings. On the sampled spectra the bound is saturated, and the measured upper edge agrees with λ_suff to within about one scan step of 0.002; equality in the thermodynamic limit is not proved. On finite rings with 0 < δ < v the spectral winding never vanishes identically; for the two ring lengths studied, the winding domains narrow rapidly above λ_suff, consistently with an estimated exponential scale. Biorthogonal entanglement entropy and polarisation equal those of the Hermitian parent. Comparison with the uniform nonreciprocal chain ties the coexistence to the energy dependence of γ_H(E) caused by dimerisation.