A Luttinger liquid is a theoretical model used in condensed matter physics to describe a one-dimensional system of interacting fermions. The model captures the behavior of fermionic particles (like electrons) in a way that accounts for their interactions, while still respecting the principles of quantum mechanics.
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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]
Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids by Duncan Haldane. Abstract: universal harmonic description and similarity of one-dimensional Bose and Fermi fluids
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.