  Equilibration of observables in closed quantum systems that are described by
a unitary time evolution is a meanwhile well-established phenomenon apart from
a few equally well-established exceptions. Here we report the surprising
theoretical observation that non-integrable spin rings with nearest neighbor
isotropic Heisenberg interaction not only equilibrate but moreover also
synchronize the directions of the expectation values of the individual spins.
In our numerical simulations, we investigate the free induction decay (FID) of
an ensemble of up to N = 25 quantum spins with s = 1/2 each by solving the
time-dependent Schr\"odinger equation numerically exactly. Our findings are
related to, but not fully explained by conservation laws of the system. The
synchronization is very robust against for instance random fluctuations of the
Heisenberg couplings and inhomogeneous magnetic fields. Synchronization is not
observed with strong enough anisotropic symmetry-breaking interactions such as
the dipolar interaction. We also compare our results to classical spin dynamics
which under all investigated conditions does not exhibit phase synchronization
due to the lack of entanglement. For classical spin systems the fixed magnitude
of individual spins effectively acts like additional N conservation laws.
