Bosonization maps 1D interacting fermions to a free boson theory:
ψ_R/L(x) = η e^{±ikFx} e^{±i(φ−θ)/√K} where φ is the boson field and θ its dual.
The Luttinger liquid (Haldane 1981) is characterized by a single parameter K:
K=1 for free fermions, K<1 for repulsive interactions, K>1 for attractive.
All correlation functions are power laws with K-dependent exponents — no quasiparticles survive.
The density-density correlator decays as ⟨ρ(x)ρ(0)⟩ ~ x^{−2} + cos(2kFx)x^{−2K}.
The single-particle spectral function has a power-law edge A(ω) ~ |ω|^α with α=(K+1/K)/2−1.
This visualization shows spatial correlation functions for various observables as a function of K.