Review of Quantum Hall Trio

The completion of quantum Hall trio, namely the quantum Hall effect (QHE), the quantum spin Hall effect (QSHE), and the quantum anomalous Hall effect (QAHE), has greatly broadened our understandings about the electron movement in solids. Likewise, the discovered scale-invariant dissipationless conduction in these three quantum transport states has inspired us to the development of practical low-power-consumption electronics/spintronics applications. In this review article, we outline the fundamental physics and relations between different Hall effects. In general, QSHE is a result of band inversion due to the large spin-orbit coupling; since the time-reversal symmetry (TRS) is well-protected in such QSH insulators, helical edge states with both spin-up and spin-down electrons are allowed and the conduction is immune to non-magnetic impurities. On the other hand, when TRS is broken by either perpendicular magnetic field or magnetic order, helical edge states are reduced to the single chiral channel, which in turn gives rise to QHE and QAHE, and backscattering from any impurity is strictly forbidden owing to the nature of chirality. Moreover, the interplay between this quantum Hall trio in certain conditions unveils new ways to explore emerging chiral materials and related quantum phenomena.