Based on a generalized free energy we derive exact thermodynamic
Bethe ansatz formulas for the expectation value of the spin current, the
spin current-charge, charge-charge correlators, and consequently the
Drude weight. These formulas agree with recent conjectures within the
generalized hydrodynamics formalism. They follow, however, directly from
a proper treatment of the operator expression of the spin current. The
result for the Drude weight is identical to the one obtained 20 years
ago based on the Kohn formula and TBA. We numerically evaluate the Drude
weight for anisotropies \Delta=\cos(\gamma)Δ=cos(γ)
with \gamma = \pi n/mγ=πn/m,
n\leq mn≤m
integer and coprime. We prove, furthermore, that the high-temperature
asymptotics for general \gamma=\pi n/mγ=πn/m—obtained
by analysis of the quantum transfer matrix eigenvalues—agrees with the
bound which has been obtained by the construction of quasi-local
charges.