The Drude weight is a central quantity for the transport properties of quantum spin chains. The canonical definition of Drude weight is directly related to Kubo formula of conductivity. However, the difficulty in the evaluation of such expression has led to several alternative formulations, accessible to different methods. In particular, the Euclidean, or imaginary-time Drude weight can be studied via rigorous renormalization group. As a result, in the past years several universality results have been proven for such quantity at zero temperature, both for integrable and non-integrable quantum spin chains. Here we establish the equivalence of the Euclidean and the canonical Drude weight at zero temperature. Our proof is based on rigorous renormalization group methods, Ward identities, and complex analytic ideas.
