small Zeeman field is required for the magnetic alignment of the Mn moments. 

Recently, tetradymite semiconductors Bi$_{2}$Te$_{3}$, Bi$_{2}$Se$_{3}$, and Sb$_{2}$Te$_{3}$ have been theoretically predicted and experimentally observed to be topological insulators with the bulk band gap as large as 0.3eV in Bi$_{2}$Se$_{3}$ $^{16–18}$. The electronic states close to the Fermi level are the bonding and anti-bonding p orbitals, and these states undergo a band inversion at the Γ point when the strength of the spin-orbit coupling is increased $^{16}$. When the thickness is reduced, the three dimensional (3D) topological insulator crosses over to a 2D topological insulator in an oscillatory fashion, as a function of the thickness $^{19}$. In this work, we predict that this family of compounds doped with proper transition metal elements (Cr or Fe) should give the QAH state, the long sought-after quantum Hall state without any external magnetic field. In sharp contrast to the ferromagnetic order in conventional dilute-magnetic-semiconductors (DMS), where free carriers are necessary to mediate the ferromagnetic couplings via the RKKY mechanism $^{20–23}$, the magnetic dopants in tetradymite compounds naturally order ferromagnetically, mediated by the large spin susceptibility (of the van Vleck type) of the host insulator state. Due to the proximity to the band inversion transition, the exchange splitting leads to a topologically non-trivial insulating band structure with a finite Chern number. The recent experimental progresses on this family of compounds have demon- strated that well-controlled layer-by-layer MBE thin film growth can be achieved $^{24–27}$, and various transition metal elements (such as Ti, V, Cr, Fe) can be substituted into the parent compounds with observable ferromagnetism even above 100K $^{28–30}$. By quantitative first- principles calculations, we further show that such ferromagnetic QAH insulating state can be also reached with Tc as high as around 70K. We will start from simple physical pictures based on model analysis, and then conclude by quantitative first-principles calculations. 

## II. THE VAN-VLECK PARAMAGNETISM IN BI$_{2}$SE$_{3}$ FAMILY COMPOUNDS 

We first discuss minimal requirements for the ferromagnetic insulator phase in a semi- conductor system doped with dilute magnetic ions under the assumption that the magnetic exchange among local moments is mediated by the band electrons. The whole system can then be divided into two sub-systems describing the local moments and band electrons re- spectively, which are coupled by a magnetic exchange term. Therefore if we only consider 

