Luttinger Liquid — Charge-Spin Separation

In a 1D Luttinger liquid, the electron fractionates into independent charge (holon) and spin (spinon) modes propagating at different velocities v_ρ ≠ v_σ. The Luttinger parameter K_ρ < 1 for repulsive interactions controls power-law decay: G(x,τ) ~ (x²+v²τ²)^−α with α = (K_ρ+1/K_ρ)/4.