Luttinger liquid

A Luttinger liquid is a low-energy description of many gapless one-dimensional interacting quantum fluids. Its natural excitations are collective density waves, rather than independent fermion-like quasiparticles. Long-distance correlations typically decay with interaction-dependent powers at zero temperature. The Luttinger parameter and a mode velocity encode much of that long-wavelength behavior.

A narrow channel containing interacting particles motivates the picture: in one dimension, density disturbances cannot spread around one another through transverse space, and collective motion becomes especially important. The description applies to suitable bosonic as well as fermionic fluids. Duncan Haldane's harmonic-fluid treatment explicitly identifies this shared low-energy structure in terms of long-wavelength density fluctuations and renormalized parameters.^[ref] A gap, strong lattice locking or additional low-energy modes can require an extended description, so one-dimensional geometry alone is not enough to establish a single Luttinger liquid theory.

The Luttinger parameter $K$ characterizes interactions and correlation exponents in a low-energy Luttinger liquid. In one common spinless convention,The parameter $v$ sets the collective-mode speed, while $K$ balances the conjugate phase and density stiffnesses. For noninteracting spinless fermions $K=1$; repulsive and attractive interactions commonly move it below and above one within this convention. Spinful models have separate charge and spin sectors. Normalization and boundary conditions must accompany comparisons between formulas.

In an appropriate spinful one-dimensional interacting conductor, low-energy density and spin disturbances form distinct collective modes, generally with different velocities. A bare injected electron can excite both modes. Boundary conditions and attached leads still matter for electrical conductance; a bulk interaction parameter alone need not set the two-terminal value of a clean contacted wire.