Directionally Independent Energy Gap Formation Due to the Hyperfine Interaction

We study energy gap formation at the level-crossing point due to the hyperfine interaction. In contrast to the energy gap induced by the Dzyaloshinskii-Moriya interaction, the gap induced by the hyperfine interaction is independent of the direction of the magnetic field. We also study the dynamics of the magnetization under a time dependent field that changes at a constant rate.