Field dependence of the magnetic order in Co$_3$V$_2$O$_8$

Co$_{3}$V$_{2}$O$_{8}$ (CVO) has a geometrically frustrated magnetic lattice, a Kagom$\acute{e}$ staircase. In zero field [1], CVO initially orders magnetically at 11.3 K into an incommensurate phase, with wave vector $k$ = (0,$\delta$,0) with $\delta$ = 0.55. $\delta$ decreases monotonically with decreasing temperature. It locks into a commensurate antiferromagnetic value of $\frac{1}{2}$ and $\frac{1}{3}$ before the ferromagnetic ground state ($\delta~=~0$) is revealed at 6.2 K. The spin direction for all spins is along the $a$ axis. A theory based on a minimal Ising model with competing exchange interactions can explain the basic features of the magnetic ordering. The application of magnetic field along the $a$ axis strongly affects all of the phases. In particular, the ferromagnetic state is suppressed in favor of the $\delta=0.5$ antiferromagnetic state. [1] Y. Chen, J. W. Lynn, Q. Huang, F. M. Woodward, T. Yildirim, G. Lawes, A. P. Ramirez, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, and A. B. Harris, Phys. Rev. B 74, 014430 (2006).