Umklapp process and resistivity in one-dimensional fermion systems

The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as e^-/T, where is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.