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Whether at zero spin density $m=0$ and finite temperatures $T>0$ the spinstiffness of the spin-$1/2$ $XXX$ chain is finite or vanishes remains anunsolved and controversial issue, as different approaches yield contradictoryresults. Here we provide an exact upper bound on the stiffness within acanonical ensemble at any fixed value of spin density $m$ and show that it isproportional to $m^2 L$ in the thermodynamic limit of chain length$L\to\infty$, for any finite, nonzero temperature. Moreover, we explicitlycompute the stiffness at $m=0$ and confirm that it vanishes. This allows us toexactly exclude the possibility of ballistic transport within the canonicalensemble for $T>0$.

Vanishing spin stiffness in the spin-1/2 Heisenberg chain for any nonzero temperature