Hatano-Nelson model: asymmetric hopping drives bulk states to boundaries
Energy spectrum — open (dots) vs periodic (ring) BC
Winding number W(E) of det[H(k)−E] around E-plane
Eigenstate amplitudes |ψ_n(x)|² (open BC)
Winding number W = 1 | Skin localisation length ξ ≈ ?
In the Hatano-Nelson model, particles hop right with amplitude t₊ and left with t₋ ≠ t₊. This asymmetry is non-Hermitian: H ≠ H†.
Under open boundary conditions (OBC), all bulk eigenstates collapse to one boundary — the skin effect. Under periodic BC (PBC), the spectrum forms a loop in the complex plane. The winding number W of this loop around a point E counts how many OBC skin modes have that energy. W = 1 here when t₊ ≠ t₋.
Localisation length: ξ = 1/|ln(t₋/t₊)|. States accumulate at the left boundary when t₊ > t₋ (rightward bias → particles pile at right edge in open chain).
Discovered in topological context by Yao & Wang (2018); the bulk-boundary correspondence must be reformulated using the non-Bloch band theory.