  Exact solutions for quantum many-body systems are rare and provide valuable
insight to universal phenomena. Here we show experimentally in anisotropic
Heisenberg chains that special helical spin patterns can have very long
lifetimes. This finding confirms the recent prediction of phantom Bethe states,
exact many-body eigenstates carrying finite momenta yet no energy. We
theoretically find analogous stable spin helices in higher dimensions and in
other non-integrable systems, where they imply non-thermalizing dynamics
associated with quantum many-body scars. We use phantom spin helices to
directly measure the interaction anisotropy which has a major contribution from
short-range off-site interactions that have not been observed before. Phantom
helix states open new opportunities for quantum simulations of spin physics and
studies of many-body dynamics.
