Kinetic Theory of Spin Diffusion and Superdiffusion in $XXZ$ Spin Chains

We address the nature of spin transport in the integrable $XXZ$ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as $D(t)\sim t^{1/3}$. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss $XXZ$ models with easy-axis anisotropy $\mathrm{\Delta }>1$. Our method gives closed-form expressions for the diffusion constant $D$ in the infinite-temperature limit for all $\mathrm{\Delta }>1$. We find that $D$ saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as $D\sim (\mathrm{\Delta }-1{)}^{-1/2}$.

Figure 1

Spin diffusion constant $D$ in the easy-axis phase $\mathrm{\Delta }>1$ as a function of $\mathrm{\Delta }$. The diffusion constant diverges in the anisotropic limit, leading to superdiffusion with $D(t)\sim t^{1/3}$. Inset: Propagation of a magnon in the large $|\mathrm{\Delta }|$, low temperature, ferromagnetic limit. The magnon is initially a minority $\downarrow$ spin in an $\uparrow$-spin domain; it is then transmitted into a $\downarrow$-spin domain as a minority $\uparrow$ spin. On average, therefore, the magnon is neutral at half-filling, giving rise to a vanishing spin Drude weight. However, its effective magnetization fluctuates, leading to diffusive spin transport coexisting with ballistic spreading of energy and, presumably, quantum information.