Domain-wall melting in the spin-$1/2$ XXZ spin chain: emergent Luttinger liquid with fractal quasi-particle charge

Understand

In spin chains with local unitary evolution preserving the magnetization $S^{\rm z}$, the domain-wall state $\left| \dots \uparrow \uparrow \uparrow \uparrow \uparrow \downarrow \downarrow \downarrow \downarrow \downarrow \dots \right>$ typically "melts".

• At large times, a non-trivial magnetization profile develops in an expanding region around the initial position of the domain-wall.
• For non-integrable dynamics the melting is diffusive, with entropy production within a melted region of size $\sqrt{t}$.
• In contrast, when the evolution is integrable, ballistic transport dominates and results in a melted region growing linearly in time, with no extensive entropy production: the spin chain remains locally in states of zero entropy at any time.