Coupling of frustrated Ising spins to magnetic cycloid in multiferroic TbMnO3

We report on diffraction measurements on multiferroic TbMnO3 which demonstrate that the Tb- and Mn-magnetic orders are coupled below the ferroelectric transition TFE = 28 K. For T < TFE the magnetic propagation vectors (tau) for Tb and Mn are locked so that tauTb = tauMn, while below TNTb = 7 K we find that tauTb and tauMn lock-in to rational values of 3/7 b* and 2/7 b*, respectively, and obey the relation 3tauTb - tauMn = 1. We explain this novel matching of wave vectors within the frustrated ANNNI model coupled to a periodic external field produced by the Mn-spin order. The tauTb = tauMn behavior is recovered when Tb magnetization is small, while the tauTb = 3/7 regime is stabilized at low temperatures by a peculiar arrangement of domain walls in the ordered state of Ising-like Tb spins.