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Using conformal field theory calculations of the energy spectrum, within theXXZ model we investigate effects of the flux insertion and the Umklapp term. Wediscuss two approaches to the evaluation of the Drude weight, the firstcorresponding to the linear response theory and the second corresponding to thetwisted boson theory with the Umklapp term. Divergences obtained in the contextof the former contradict the Bethe ansatz results, with the two approachescoinciding for the free fermion point only. The origin of this discrepancy isin the different order in which the Umklapp term and the flux insertion aretreated, where the marginal perturbation should be considered before theirrelevant. We calculate the scaling of the conductivity with system size andtemperature in the long-wave limit.

Kubo's response theory and bosonization with a background gauge field and irrelevant perturbations