The very-low-temperature thermal conductivity (kappa) is studied for BaCo_2V_2O_8, a quasi-one-dimensional Ising-like antiferromagnet exhibiting an unusual magnetic-field-induced order-to-disorder transition. The nearly isotropic transport in the longitudinal field indicates that the magnetic excitations scatter phonons rather than conduct heat. The field dependence of kappa shows a sudden drop at sim 4 T, where the system unndergoes the transition from the Neel order to the incommensurate state. Another dip at lower field of sim 3 T indicates an unknown magnetic transition, which is likely due to the spin-flop transition. Moreover, the kappa(H) in the transverse field shows a very deep valley-like feature, which moves slightly to higher field and becomes sharper upon lowering the temperature. This indicates a magnetic transition induced by the transverse field, which however is not predicted by the present theories for this low-dimensional spin system.
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