We report a heat switch effect in single crystals of an antiferromagnet Co$_3$V$_2$O$_8$, that is, the thermal conductivity ($kappa$) can be changed with magnetic field in an extremely large scale. Due to successive magnetic phase transitions at 12--6 K, the zero-field $kappa(T)$ displays a deep minimum at 6.7 K and rather small magnitude at low temperatures. Both the temperature and field dependencies of $kappa$ demonstrate that the phonons are strongly scattered at the regime of magnetic phase transitions. Magnetic field can suppress magnetic scattering effect and significantly recover the phonon thermal conductivity. In particular, a 14 T field along the $a$ axis increases the $kappa$ at 7.5 K up to 100 times. For $H parallel c$, the magnitude of $kappa$ can be suppressed down to $sim$ 8% at some field-induced transition and can be enhanced up to 20 times at 14 T. The present results demonstrate that it is possible to design a kind of heat switch in the family of magnetic materials.
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