  We calculate the magnetic response of a buckled honeycomb lattice with
intrinsic spin-orbit coupling (such as silicene) which supports valley-spin
polarized energy bands when subjected to a perpendicular electric field $E_z$.
By changing the magnitude of the external electric field, the size of the two
band gaps involved can be tuned, and a transition from a topological insulator
(TI) to a trivial band insulator (BI) is induced as one of the gaps becomes
zero, and the system enters a valley-spin polarized metallic state (VSPM). In
an external magnetic field ($B$), a distinct signature of the transition is
seen in the derivative of the magnetization with respect to chemical potential
($\mu$) which gives the quantization of the Hall plateaus through the Streda
relation. When plotted as a function of the external electric field, the
magnetization has an abrupt change in slope at its minimum which signals the
VSPM state. The magnetic susceptibility ($\chi$) shows jumps as a function of
$\mu$ when a band gap is crossed which provides a measure of the gaps'
variation as a function of external electric field. Alternatively, at fixed
$\mu$, the susceptibility displays an increasingly large diamagnetic response
as the electric field approaches the critical value of the VSPM phase. In the
VSPM state, magnetic oscillations exist for any value of chemical potential
while for the TI, and BI state, $\mu$ must be larger than the minimum gap in
the system. When $\mu$ is larger than both gaps, there are two fundamental
cyclotron frequencies (which can also be tuned by $E_z$) involved in the
de-Haas van-Alphen oscillations which are close in magnitude. This causes a
prominent beating pattern to emerge.
