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 (τ) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy $\left|\Delta \right|<1$. 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 $\left|\Delta \right|<1$ the decay for long time t is nowhere continuous as a function of anisotropy: it is Gaussian at roots of unity and exponential otherwise. We also determine that the front moves as $x_f\left(t\right)=t\sqrt{1-{\Delta }^2}$, by the analytic continuation of known arctic curves in the six-vertex model. Exactly at $\left|\Delta \right|=1$, we find the return probability decays as $e^{-\zeta \left(3/2\right)\sqrt{t/\pi }}{t}^{1/2}O\left(1\right)$. 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. (paper: quantum statistical physics, condensed matter, integrable systems)