Multi-$q$ crystal and magnetic structure in TbMnO$_{3}$: Evidence for a Soliton-lattice

In TbMnO$_{3}$, Mn-spins order with a sinusoidal antiferromagnetic (AF) propagation wave vector $Q_{Mn}$=[0,k+/-q,l] ($q\sim0.288b^{\ast}$) at T$_{N}$(Mn)=41K. The propagation vector $Q_{Mn}$ varies with temperature on cooling until T$_{Lock}$(Mn) $\sim$30K, which coincides with a ferroelectric transition. In addition to $Q_{Mn}$ reflections we find magnetic reflections at 3$Q_{Mn}$. Our X-ray measurements show that the magneto-elastic coupling gives rise to a structural modulation at twice the magnetic wavevector (2$Q_{Mn}$). Field cooling the sample under a magnetic field oriented along the a-direction with H$>$9T shows that all magnetic wavevectors for Mn and Tb collapse to a single $q$ structure with $Q$=[0,1/4,0], an up-up, down-down phase, and coincide with anomalies in the polarization. We argue that the temperature and field dependence of the magnetic and superlattice reflections are consistent with a soliton formalism which predicts a stable commensurate single q=1/4 phase [1]. [1] Kimura {\sl et al.}, $PRB$ 68, 60403(2003).