One-dimensional bosonization and Luttinger liquids

For a clean single-channel spinless system with two Fermi points, resolve the electron field into slowly varying right- and left-moving parts multiplied by their rapid Fermi phases. Linearize the dispersion close to those points and retain smooth forward interactions. The cutoff excludes momenta for which band curvature or additional channels matter. Density fluctuations then obey a closed bosonic algebra, allowing a quadratic field Hamiltonian for the gapless Tomonaga–Luttinger fixed-point theory.

Choose fields phi and theta so that the smooth density change is minus the spatial derivative of phi divided by pi. Their conjugacy fixes the relative normalization of the two stiffness terms. The velocity v controls propagation, while the dimensionless Luttinger parameter K controls interaction-dependent correlations. In the spinless convention K equals one for free fermions; common repulsive forward interactions give K below one. A numerical K quoted without a field or spin convention is not sufficient to reconstruct a correlation exponent.

A fermion operator becomes an exponential of bosonic fields. Because the fields fluctuate logarithmically in a gapless one-dimensional theory, the correlator of their exponentials becomes a power law. Its exponent depends on both K and its inverse: modifying one density stiffness also changes its conjugate phase fluctuations. The free value recovers the usual inverse-distance equal-time fermion correlator, while interactions generally make the electron correlator decay with a nontrivial exponent.

This power-law structure produces a continuum or branch-cut electron response rather than an isolated Landau electron pole. A vanishing quasiparticle residue in the thermodynamic low-energy limit does not mean that spectral normalization fails or that charge is lost. The formula applies within the fixed-point regime and to a specified bulk correlator; boundary tunneling has different exponents. Band curvature, finite size, finite temperature and additional modes cut off the ideal asymptotic behavior in an experiment.

For spinful electrons, appropriate low-energy regimes separate collective charge and spin fields with their own velocities and interaction parameters. A localized perturbation can then generate spin and charge responses that arrive at different times. This is spin-charge separation of collective excitations in a specified one-dimensional phase, rather than a statement that an electron visibly splits into two classical particles. Couplings between sectors and irrelevant operators can modify finite-energy behavior even when the low-energy separation is useful.

Commensurate umklapp scattering, attractive backscattering, disorder or interchain coupling can open gaps or destabilize the gapless theory. In particular the half-filled repulsive Hubbard chain has a gapped charge sector, so the fully gapless two-sector description cannot be applied there without qualification. Model solutions and momentum-resolved spectra provide specific tests; the existence of a power law in an unrelated measurement is not by itself proof of fractionalization. The wave lab has one common fixed frequency and cannot simulate different spin and charge velocities.