Coupling of frustrated ising spins to the magnetic cycloid in multiferroic TbMnO(3).

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