Luttinger Liquid & Bosonization

Luttinger liquid (Tomonaga 1950, Luttinger 1963, solved by Mattis-Lieb) is the universal fixed point for interacting 1D metals. Bosonization maps fermions to bosons: ψ† ~ e^{iφ}, with φ a bosonic field. The exactly solvable Hamiltonian: H = v/2·∫dx[(∇φ)²/K + K(∇θ)²] where K is the Luttinger parameter encoding interactions. Correlations: ⟨c†(r)c(0)⟩ ~ r^{-(1+α)} with α=(K+K⁻¹-2)/4. Spin-charge separation: spinons and holons propagate at different velocities (v_s ≠ v_c). No Fermi liquid quasiparticles — the momentum distribution n(k) has a power-law singularity (not a jump) at k_F.