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Relaxation in Luttinger liquids: Bose-Fermi duality

We explore the life time of excitations in a dispersive Luttinger liquid. We perform a bosonization supplemented by a sequence of unitary transformations that allows us to treat the problem in terms of weakly interacting quasiparticles. The relaxation described by the resulting Hamiltonian is analyzed by bosonic and (after a refermionization) by fermionic perturbation theory. We show that the the fermionic and bosonic formulations of the problem exhibit a remarkable strong-weak-coupling duality. Specifically, the fermionic theory is characterized by a dimensionless coupling constant $λ= m^*l^2T$ and the bosonic theory by $λ^{-1}$, where $1/m^*$ and $l$ characterize the curvature of the fermionic and bosonic spectra, respectively, and $T$ is the temperature.