Exceptional Point — Non-Hermitian Degeneracy
At an exceptional point (EP), two eigenvalues and their eigenvectors coalesce — a fundamentally different degeneracy from Hermitian systems with distinct eigenstates.
PT-Symmetric Non-Hermitian System: H = [[δ+iγ, κ],[κ, −δ−iγ]] (gain-loss coupler). Eigenvalues E± = ±√(κ²−γ²+δ²+2iδγ). At κ=γ (EP), eigenvalues and eigenvectors coalesce — unlike Hermitian degeneracy, the Jordan block is defective (only one eigenvector). The Riemann sheet topology near an EP is a branch point with square-root winding: encircling the EP swaps eigenvalues and returns after two loops. In PT-symmetric phase (γ < κ): real spectrum. Broken phase (γ > κ): complex eigenvalues (amplified/damped modes).