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Whether in the thermodynamic limit, vanishing magnetic field $h\rightarrow0$, and nonzero temperature the spin stiffness of the spin-$1/2$ $XXX$Heisenberg chain is finite or vanishes within the grand-canonical ensembleremains an unsolved and controversial issue, as different approaches yieldcontradictory results. Here we provide an upper bound on the stiffness and showthat within that ensemble it vanishes for $h\rightarrow 0$ in the thermodynamiclimit of chain length $L\to\infty$, at high temperatures $T\rightarrow\infty$.Our approach uses a representation in terms of the $L$ physical spins $1/2$.For all configurations that generate the exact spin-$S$ energy and momentumeigenstates such a configuration involves a number $2S$ of unpaired spins $1/2$in multiplet configurations and $L-2S$ spins $1/2$ that are paired within$M_{\rm sp}=L/2-S$ spin-singlet pairs. The Bethe-ansatz strings of length $n=1$and $n>1$ describe a single unbound spin-singlet pair and a configurationwithin which $n$ pairs are bound, respectively. In the case of $n>1$ pairs thisholds both for ideal and deformed strings associated with $n$ complexrapidities with the same real part. The use of such a spin $1/2$ representationprovides useful physical information on the problem under investigation incontrast to often less controllable numerical studies. Our results providestrong evidence for the absence of ballistic transport in the spin-$1/2$ $XXX$Heisenberg chain in the thermodynamic limit, for high temperatures$T\to\infty$, vanishing magnetic field $h\rightarrow 0$ and within thegrand-canonical ensemble.

Absence of high-temperature ballistic transport in the spin-$1/2$ $XXX$ chain within the grand-canonical ensemble