Non-Hermitian Skin Effect & Winding Number

Non-Hermitian skin effect (NHSE): in systems with asymmetric hopping (t_R ≠ t_L), virtually all bulk eigenstates collapse to one boundary under OBC — a macroscopic breakdown of bulk-boundary correspondence for Bloch bands. Under PBC the spectrum traces a loop in ℂ; the winding number W = (1/2πi)∮ d[log E(k)] counts how many times the spectrum winds around any reference point inside the loop. W≠0 signals NHSE. The localization length ξ = 1/|ln(t_R/t_L)| characterizes the exponential decay. Non-Bloch band theory replaces e^{ik} with β (complex) to correctly predict the OBC spectrum.