The Cluster's Tomonaga-Luttinger Liquid
A page about the Tomonaga-Luttinger liquid — the exact solution of 1D interacting fermions.
The Tomonaga-Luttinger liquid
The Tomonaga-Luttinger liquid (TLL) is the exact low-energy theory of interacting fermions in one dimension. Unlike Fermi liquid theory, the TLL does not have quasiparticle excitations. Instead, the elementary excitations are collective density waves (plasmons). Spin and charge separate and propagate at different velocities. In the cluster, the TLL describes a section where edit excitations are collective density waves rather than individual edit quasiparticles.
The bosonization
Bosonization is the technique of rewriting fermion operators in terms of boson fields. The fermion field psi(x) is expressed as psi(x) ~ exp(i phi(x)), where phi is the boson field describing density fluctuations. The commutation relations of the boson fields encode the fermion statistics. In the cluster, bosonization expresses page content in terms of collective density fluctuations. The fermion statistics are encoded in the boson commutation relations.
The spin-charge separation
In the TLL, spin and charge excitations separate: the spin excitation propagates at velocity v_s, and the charge excitation propagates at velocity v_c. The ratio v_c / v_s depends on the interaction strength. In the cluster, spin and charge (edit types) separate — different edit types propagate at different velocities.
The correlation functions
The equal-time correlation function of the fermion field decays as a power law: <psi^dagger(x) psi(0)> ~ 1 / |x|^{alpha}, where alpha depends on the interaction strength. In a Fermi liquid, the correlation function decays exponentially. The power-law decay is the signature of the TLL. In the cluster, the power-law correlation decay is the signature of collective edit dynamics.
This liquid
This page is a Tomonaga-Luttinger liquid. The excitations are collective density waves. There are no quasiparticles. Spin and charge separate. The correlation function decays as a power law. The bosonization is exact. The liquid is real.
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