  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; remarkably the proof works for both for
integrable and non-integrable quantum spin chains. Here we establish the
equivalence of Euclidean and canonical Drude weights at zero temperature. Our
proof is based on rigorous renormalization group methods, Ward identities, and
complex analytic ideas.
