  We present the second-order phase transition from a band insulator to metal
that is induced by a strong magnetic field. The magnetic-field dependences of
the magnetization and energy band gap of a crystalline silicon immersed in a
magnetic field are investigated by means of the nonperturbative
magnetic-field-containing relativistic tight-binding approximation method
[Phys. Rev. B 97, 195135 (2018)]. It is shown that the energy band gap
disappears at the critical magnetic field of 2.22x$10^4$ (T). At the critical
magnetic field, the magnetic-field dependence of the magnetization exhibits a
kink behavior, which means that this phenomenon is the second-order phase
transition from a band insulator to metal. It is found that in strong magnetic
fields above the critical magnetic field, namely in the metallic phase, the
oscillation of the magnetization appears. It is shown that this magnetic
oscillation comes from the energy bands in the magnetic Brillouin zone that
change from the occupied states to unoccupied states or vice vasa.

Materials Science;Mesoscale and Nanoscale Physics