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 ∣Δ∣<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 τ2, which reflects the fact that the model exhibits
the limit shape phenomenon. In real time, we observe that in the region
∣Δ∣<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 xf(t)=t1−Δ2, by analytic continuation of known arctic curves in
the six vertex model. Exactly at ∣Δ∣=1, we find the return probability
decays as e−ζ(3/2)t/πt1/2O(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.Comment: 33 pages, 8 figures. v2: typos fixed, references added. v3: minor
change