Drude Weight in Non Solvable Quantum Spin Chains

For a quantum spin chain or 1D fermionic system, we prove that the Drude weight D verifies the universal Luttinger liquid relation nu(2)(s) = D/kappa where. is the susceptibility and vs is the Fermi velocity. This result is proved by rigorous Renormalization Group methods and is true for any weakly interacting system, regardless its integrability. This paper, combined with Benfatto and Mastropietro ( in J. Stat. Phys. 138, 1084-1108, 2010), completes the proof of the Luttinger liquid conjecture for such systems.