integral over the MBZ gives the topological invariant, namely, the Chern number C$_{n}$ = − 1 2 π ∫ $_{MBZ}$d $^{2}$k Ω$_{n}$( k ). The sum of Chern numbers over the integer occupied bands gives the quantized Hall conductance in unit of e $^{2}$/h (see numbers in Fig. 2). µ in Eq. (8) is the electron chemical potential, β =1 /k$_{B}$T , and ϵ$_{n}$$_{k}$ is the magnetic band energy. When the field varies by an infinitesimal quantity, i.e., from B to B + δ B , then the magnetic band energy can be linearly expanded as ϵ$_{n}$$_{k}$ ( B + δ B ) = ϵ$_{n}$$_{k}$ ( B ) − m$_{n}$$_{k}$ ( B ) · δ B , where m$_{n}$$_{k}$ is the conventional crystal orbital magnetic moment defined by m$_{n}$$_{k}$ = − i ( e/ 2 ℏ ) 〈∇ $_{k}$u$_{n}$$_{k}$ | × [ H ( k ) − ϵ$_{n}$$_{k}$ ] |∇ $_{k}$u$_{n}$$_{k}$ 〉 in the MBZ. The magnetization is then given by the field derivative at fixed temperature and chemical potential, M = − ( ∂F/∂ B )$_{µ,T}$ , with the result where f$_{n}$ ( k ) is the equilibrium Fermi-Dirac distribution function for n th magnetic Bloch band. The first term in Eq. (9) is just a statistical sum of the orbital magnetic moments of the carriers originating from the self-rotation of the carrier wave packet [19, 20], thus we call this term the conventional part M$_{c}$ of the orbital magnetization. Whereas the second term M$_{Ω}$ is the Berry-phase correction to the orbital magnetization. This term is of topological nature. Interestingly, it is this Berry-phase term that eventually enters the transport current [21]. At zero temperature the general expression (9) reduces to [10, 16, 22] where the upper limit means that the integral is over magnetic Bloch states with energies below the Fermi energy µ$_{0}$ . Now with the help of the knowledge on the magnetic energy spectrum obtained from diagonalizing the Hamiltonian (7), we can numerically obtain the magnetization M for different field f . First, let us consider the case with field f =1 / 4. In this case, the energy spectrum splits into three bands between which there are two band gaps with the same gap width Δ=1.74 (see Fig. 3(a)). The middle band collapse into a plane in the MBZ. The calculated zero-temperature magnetization M is plotted in Fig. 3(b). From this figure, one can find that with the Fermi energy µ$_{0}$ increases from − 3.26, the lower band begins to be M = ∑ n ∫ µ 0 MBZ d $^{2}$k [ m$_{n}$ ( k ) + e $_{ℏ}$Ω$_{n}$ ( k )( µ$_{0}$ − ϵ$_{n}$$_{k}$ ) ] , (10) M = ∑ n ∫$_{MBZ}$ d $^{2}$km$_{n}$ ( k ) f$_{n}$ ( k ) (9) + 1 β ∑ n ∫$_{MBZ}$ d $^{2}$k e $_{ℏ}$Ω$_{n}$ ln [ 1 + e β ( µ − ϵ$_{n}$$_{k}$ $^{)}$] ≡ M$_{c}$ + M$_{Ω}$ , 7