Spin dynamics in Kagom\'{e}-staircase multiferroic Ni$_3$V$_2$O$_8$

The coupling of magnetic and ferroelectric order has recently been drawing a lot of interest in condensed matter science given the fundamental interest and potential applications. Ni$_3$V$_2$O$_8$ (NVO) is a $S=1$ magnet with Ni$^{2+}$ ions arranged in a weakly coupled buckled Kagom\'{e}-staircase planes. Its complex magnetic phase diagram exhibits four different zero field incommensurate and commensurate magnetic phases below 10~K, with only one developing ferroelectric order. We present here a detailed study of low temperature magnetic dynamics in this geometrically frustrated spin system. Using single crystal inelastic neutron scattering technique we map the magnetic excitation spectra across all the magnetic phase transitions. We found that the spin-waves, well formed in the base temperature nonferroelectric phase at $T < 3$~K, are considerably damped when the system enters the low-temperature incommensurate phase with ferroelectric order ($3.9 < T < 6.3$~K). Finally, we discuss models that describe the coupling between magnetic and ferroelectric properties in the incommensurate magnets.