Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

We study the return probability and its imaginary ($\tau$) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }<1$. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to ${\tau }^2$, which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }<1$ the decay for large times $t$ is nowhere continuous as a function of anisotropy: it is either gaussian at root of unity or exponential otherwise. As an aside, we also determine that the front moves as $x_{\rm f}(t)=t\sqrt{1-\Delta^2}$, by analytic continuation of known arctic curves in the six vertex model. Exactly at $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }=1$, we find the return probability decays as $e^{-\zeta (3\mathrm{/}2)\sqrt{t\mathrm{/}\pi }}{t}^{1\mathrm{/}2}O(1)$. It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench.