Asymmetric hopping (t_R ≠ t_L) causes ALL bulk eigenstates to pile up at one edge under open boundary conditions. PBC spectrum forms a loop in ℂ; OBC eigenvalues collapse inside it.
Parameters
Non-Hermitian ratio
—
Skin length ξ
—
Edge localizing
—
H = Σ[t_R|n+1⟩⟨n| + t_L|n⟩⟨n+1|]
PBC: E(k) = t_R e^{ik} + t_L e^{-ik} → loop
OBC: all states ∝ (t_L/t_R)^n → skin
Localization length: ξ = 1/ln(t_R/t_L)
Point gap (PBC loop) → topological origin