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

Abstract

Whether in the thermodynamic limit, vanishing magnetic field h →0, and nonzero temperature the spin stiffness of the spin-1/2XXXHeisenberg chain is finite or vanishes within the grand-canonical ensemble remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an upper bound on the stiffness and show that within that ensemble it vanishes for h →0in the thermodynamic limit of chain length L →∞, at high temperatures T→∞. Our approach uses a represen-tation in terms of the Lphysical spins 1/2. For all configurations that generate the exact spin-Senergy and momentum eigenstates such a configuration involves a number 2Sof unpaired spins 1/2in multiplet con-figurations and L −2Sspins 1/2that are paired within Msp=L/2 −Sspin–singlet pairs. The Bethe-ansatz strings of length n =1and n >1describe a single unbound spin–singlet pair and a configuration within which npairs are bound, respectively. In the case of n >1pairs this holds both for ideal and deformed strings associated with ncomplex rapidities with the same real part. The use of such a spin 1/2repre-sentation provides useful physical information on the problem under investigation in contrast to often less controllable numerical studies. Our results provide strong evidence for the absence of ballistic transport in the spin-1/2XXXHeisenberg chain in the thermodynamic limit, for high temperatures T→∞, vanishing magnetic field h →0and within the grand-canonical ensemble.