Critical points where ∇_k E(k) = 0 create divergences in the DOS — logarithmic in 2D, inverse-square-root in 1D, and cusps in 3D
Band Structure
Dimension & View
Van Hove singularities arise at high-symmetry k-points (Γ, M, X) where the group velocity vanishes. In 2D square lattice: E(k) = −2t(cos kx + cos ky) − 4t' cos kx cos ky. The DOS g(E) = (1/2π²) ∮ dk/|∇E| diverges logarithmically at the saddle point E = 0 (band center). This nesting at half-filling enhances susceptibility χ(q), driving instabilities: CDW, spin-density wave, or superconductivity. The 1D DOS diverges as (E−E₀)^(−1/2) at band edges — the classic inverse-square-root.