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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 profiledevelops in an expanding region around the initial position of the domain-wall.For non-integrable dynamics the melting is diffusive, with entropy productionwithin a melted region of size $\sqrt{t}$. In contrast, when the evolution isintegrable, ballistic transport dominates and results in a melted regiongrowing linearly in time, with no extensive entropy production: the spin chainremains locally in states of zero entropy at any time. Here we show that, forthe integrable spin-$1/2$ XXZ chain, low-energy quantum fluctuations in themelted region give rise to an emergent Luttinger liquid which, remarkably,differs from the equilibrium one. The striking feature of this emergentLuttinger liquid is its quasi-particle charge (or Luttinger parameter $K$)which acquires a fractal dependence on the XXZ chain anisotropy parameter$\Delta$.

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