Finite-temperature Transport Properties of a Spin Quantum System

The gapped spin- $${1\over 2}$$ XXZ chain with spin anisotropy $$\Delta > 1$$ is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when $$\Delta > 1$$ , and diverges in the limit $$\Delta \rightarrow 1$$ . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for $$\Delta > 1$$ , but becomes superdiffusive as $$\Delta \rightarrow 1$$ . These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.