spin-orbital coupling with band inversion, therefore the matrix element of the spin operator can be very large. The Van-Vleck paramagnetism in Bi$_{2}$Se$_{3}$ family can be well understood using the follow- ing effective k · p Hamiltonian derived in ref. $^{16}$, with k$_{±}$ = k$_{x}$ ± ik$_{y}$ , ϵ$_{0}$ ( k ) = C + D$_{xy}$ ( k 2 x + k 2 y ) + D$_{z}$k 2 $_{z}$, M ( k ) = M$_{0}$ + B$_{xy}$ ( k 2 x + k 2 y ) + B$_{z}$k 2 z $^{32}$, in the basis of | P + z , ↑〉 , | P − z , ↓〉 , | P + z , ↓〉 , | P − z , ↑〉 . The ± in the basis denote the even and odd parity states and ↓↑ indicate the spin. As discussed before in Ref. $^{16}$, the four low energy states originate from the bonding and anti-bonding combinations of Bi and Se P$_{z}$ orbitals, and the parity is a good quantum number at the Γ point due to the spatial inversion symmetry. The topological properties of the Bi$_{2}$Se$_{3}$ family can be well explained by the band inversion between two sets of Kramer’s degenerate states with opposite parity at the Γ point. Here we will show that the band inversion not only gives the non-trivial topological index but also increases the spin susceptibility dramatically. For simplicity, we focus on a 2D sheet in the k-space with k$_{z}$ = 0 (the situation of general k$_{z}$ is qualitatively similar), where the above effective Hamiltonian reduces to two diagonal blocks as 10,19 with h ( k ) = M ( k ) σ$_{z}$ + A$_{xy}$ ( k$_{x}$σ$_{x}$ + k$_{y}$σ$_{y}$ ) + ϵ$_{0}$ ( k ). The two eigen-states of h ( k ) give the occupied and unoccupied bands respectively, which can be denoted as | ck 〉 and | vk 〉 . It is then obvious that the amplitude of the matrix element 〈 ck | ˆ S$_{z}$ | vk 〉 reaches the maximum value when M ( k ) = 0. The above condition can only be satisfied when M$_{0}$B$_{xy}$ < 0, which is exactly the condition for the band inversion. In the normal phase without inverted bands, the maximum value of 〈 ck | ˆ S$_{z}$ | vk 〉 can not be reached and the spin susceptibility is less pronounced compared to the system with band inversion. To further demonstrate this argument, we perform first-principles calculations (see ref. 33 for method) for the electronic structures and the spin susceptibility of Bi$_{2}$Se$_{3}$ bulk. The results as function of SOC strength are plotted in Fig. 1(a). When the relative SOC strength H$_{3}$$_{D}$ ( k ) =     M ( k ) A$_{xy}$k$_{−}$ 0 A$_{z}$k$_{z}$ A$_{xy}$k$_{+}$ −M ( k ) A$_{z}$k$_{z}$ 0 0 A$_{z}$k$_{z}$ M ( k ) − A$_{xy}$k$_{+}$ A$_{z}$k$_{z}$ 0 − A$_{xy}$k$_{−}$ −M ( k )    $^{+}$$^{ϵ$_{0}$}$$^{(}$$^{k}$$^{)}$ (2) H$_{2}$$_{D}$ ( k ) =   $^{h}$$^{(}$$^{k}$$^{)}$ 0 0 h $^{∗}$( − k )    (3) 6