  We investigate the entanglement spreading in the anisotropic spin-1/2
Heisenberg (XXZ) chain after a geometric quench. This corresponds to a sudden
change of the geometry of the chain or, in the equivalent language of
interacting fermions confined in a box trap, to a sudden increase of the trap
size. The entanglement dynamics after the quench is associated with the
ballistic propagation of a magnetization wavefront. At the free fermion point
(XX chain), the von-Neumann entropy S_A exhibits several intriguing dynamical
regimes. Specifically, at short times a logarithmic increase is observed,
similar to local quenches. This is accurately described by an analytic formula
that we derive from heuristic arguments. At intermediate times partial revivals
of the short-time dynamics are superposed with a power-law increase S_A
t^\alpha, with \alpha<1. Finally, at very long times a steady state develops
with constant entanglement entropy. As expected, since the model is integrable,
we find that the steady state is non thermal, although it exhibits extensive
entanglement entropy. We also investigate the entanglement dynamics after the
quench from a finite to the infinite chain (sudden expansion). While at long
times the entanglement vanishes, we demonstrate that its relaxation dynamics
exhibits a number of scaling properties. Finally, we discuss the short-time
entanglement dynamics in the XXZ chain in the gapless phase. The same formula
that describes the time dependence for the XX chain remains valid in the whole
gapless phase.
