Transport in the $\mathrm{XX}$ chain at zero temperature: Emergence of flat magnetization profiles

We study the connection between magnetization transport and magnetization profiles in zero-temperature $\mathrm{XX}$ chains. The time evolution of the transverse magnetization $m(x,t)\mathrm{}$ is calculated using an inhomogeneous initial state that is the ground state at fixed magnetization but with m reversed from $-m_0$ for $x<\mathrm{}0\mathrm{}$ to $m_0$ for $x>\mathrm{}0.\mathrm{}$ In the long-time limit, the magnetization evolves into a scaling form $m(x,t)=\Phi \mathrm{}(x/t)\mathrm{}$ and the profile develops a flat part $(m=\Phi =0)$ in the $|x/t|<~{c(m}_0)$ region. The flat region shrinks to zero if $m_0\to 1/2$ while it expands with the maximum velocity $c_0=1\mathrm{}$ for $m_0\to 0.$ The states emerging in the scaling limit are compared to those of a homogeneous system where the same magnetization current is driven by a bulk field, and we find that the expectation values of various quantities (energy, occupation number in the fermionic representation) agree in the two systems.