content_image/371627.jpg content_image/371628.jpg content_image/371629.jpg content_image/371630.jpg content_image/371631.jpg
λ/λ$_{0}$ is larger than 0.5, the bands at the Γ point are inverted, and the spin susceptibility starts to increase appreciably around this point. 

## III. MEAN FIELD THEORY FOR THE MAGNETIC TOPOLOGICAL INSULA- TORS 

In the following, we apply a tight binding version of the Zener model 20–22 to calculate the possible Curie temperature at the mean field level. Let’s first assume that some magnetic local moments are introduced iso-electronically into the Bi$_{2}$Se$_{3}$ compounds without changing the insulating behavior. We show that ferromagnetic ordering with T$_{c}$ as high as 70K can be achieved via the van-Vleck paramagnetism, without requiring the existence of free carriers as in conventional DMS systems. The total Hamiltonian of our system can be written as 

<img src='content_image/371627.jpg'>

with 

<img src='content_image/371628.jpg'>

is the single particle Hamiltonian (obtained from first-principles calculations) expressed in the basis of local Wannier functions $^{33}$. Here i and j denote the sites, I only runs over all the unit cells containing magnetic ions and µν are the orbital/spin indices. The coupling strength between the local spin S$_{I}$ and the band spin s$_{µν}$ of band electrons is described by the effective exchange parameter J$_{eff}$ . 

Performing the mean field approximation and applying the virtual crystal approximation (CPA) to average out the spatial inhomogeneity, Eq.(4) can be written as 

<img src='content_image/371629.jpg'>

where M α L = 〈 S α I $_{=0}$〉 , M α e = 〈 s α $_{µν}$c † 0 $_{,µ}$c$_{0}$$_{,ν}$ 〉 are the mean field magnetization for the local moments and band electrons respectively, with 

and 

<img src='content_image/371630.jpg'>

<img src='content_image/371631.jpg'>

