  The relativistic Dirac equation covers the fundamentals of electronic
phenomena in solids and as such it effectively describes the electronic states
of the topological insulators like Bi$_2$Se$_3$ and Bi$_2$Te$_3$. Topological
insulators feature gapless surface states and, moreover, magnetic doping and
resultant ferromagnetic ordering break time-reversal symmetry to realize
quantum anomalous Hall and Chern insulators. Here we focus on the bulk and
investigate the mutual coupling of electronic and magnetic properties of Dirac
electrons. Without carrier doping, spiral magnetic orders cause a ferroelectric
polarization through the spin-orbit coupling. In a doped metallic state, the
anisotropic magnetoresistance arises without uniform magnetization. We find
that electric current induces uniform magnetization and conversely an
oscillating magnetic order induces electric current. Our model provides a
coherent and unified description of all those phenomena. The mutual control of
electric and magnetic properties demonstrates implementations of
antiferromagnetic spintronics. We also discuss the stoichiometric magnetic
topological insulator MnBi$_2$Te$_4$.
