  Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb
lattice, which shares almost every remarkable property with graphene. The low
energy structure of silicene is described by Dirac electrons with relatively
large spin-orbit interactions due to its buckled structure. The key observation
is that the band structure is controllable by applying the electric field to a
silicene sheet. We explore the phase diagram of silicene together with exchange
field $M$ and by applying electric field $E_{z}$. There appear quantum
anomalous Hall (QAH) insulator, valley polarized metal (VPM), marginal valley
polarized metal (M-VPM), quantum spin Hall (QSH) insulator and band insulator
(BI). They are characterized by the Chern numbers and/or by the edge modes of a
nanoribbon. It is intriguing that electrons have been moved from a conduction
band at the K point to a valence band at the K' point for $E_{z}>0$ in the VPM.
We find in the QAH phase that flat gapless edge modes emerge and that spins
form a momentum-space skyrmion to yield the Chern number. It is remarkable that
a topological quantum phase transition can be induced simply by changing
electric field in a single silicene sheet.
