  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. Here we show that, for
the integrable spin-$1/2$ XXZ chain, low-energy quantum fluctuations in the
melted region give rise to an emergent Luttinger liquid which, remarkably,
differs from the equilibrium one. The striking feature of this emergent
Luttinger liquid is its quasi-particle charge (or Luttinger parameter $K$)
which acquires a fractal dependence on the XXZ chain anisotropy parameter
$\Delta$.
