Magnetic Phase Diagram of Co$_{3}$V$_{2}$O$_{8}$

Kagom\'{e}-staircase lattice structures like Ni$_{3}$V$_{2}$O$_{8}$ and Co$_{3}$V$_{2}$O$_{8}$ have recently attracted attention because of their complex magnetic phase diagrams and the magnetically induced ferroelectric (FE) phase observed in Ni$_{3}$V$_{2}$O$_{8}$. Co$_{3}$V$_{2}$O$_{8}$ at zero magnetic field exhibits five subsequent magnetic phase transition in a narrow temperature range. It has an incommensurate antiferromagnetic phase below T$_{N}$=11.4 K and weak ferromagnetic behavior along the a-axis at T$_{C}$=6.2 K. Along with three other phase transitions in between; T$_{1}$=8.9 K, T$_{2}$=7.0 K and T$_{3}$=6.9 K, the evolution of these five phase transitions under magnetic field, phase boundaries, is traced through magnetic susceptibility and dielectric constant anomalies. We resolve the complete magnetic phase diagram of Co$_{3}$V$_{2}$O$_{8}$ with the magnetic field applied along the principal crystallographic orientations.