How can I derive the matrix value of bond operator? For SU(2), if bond operator is 
how can I derive the matrix of bond operator nuder the basis of spin coherent state:

 I just know I should use the overlap relation of spin coherent state, but I don't know how to deal with the bond operator.
 A: I am following section 7.3 in Interacting electrons and quantum magnetism by Assa Auerbach.
The spin coherent states in the Schwinger boson representation are given by:
$$ | \Omega \rangle_S = e^{iS\chi} \frac{ (u a^{\dagger}+ v b^{\dagger})^{2S}}{\sqrt{(2S)!}} | 0\rangle$$
The annihilation operator $a$ acts as a derivative with respect to $a^{\dagger}$, thus:
$$ a | \Omega \rangle_S = e^{iS\chi} u \frac{ (u a^{\dagger}+ v b^{\dagger})^{2S-1}}{\sqrt{(2S)!}} | 0\rangle = \sqrt{2S} e^{i\frac{\chi}{2}} u| \Omega \rangle_{S-\frac{1}{2}}$$
Similarly:
$$ b | \Omega \rangle_S =\sqrt{2S} e^{i\frac{\chi}{2}} v| \Omega \rangle_{S-\frac{1}{2}}$$
$$ \langle \Omega|_S  a^{\dagger}  = \sqrt{2S} e^{i\frac{\chi}{2}} u^{*} \langle \Omega|_{S-\frac{1}{2}} $$
and
$$ \langle \Omega|_S  b^{\dagger}  = \sqrt{2S} e^{i\frac{\chi}{2}} v^{*} \langle \Omega|_{S-\frac{1}{2}} $$
Performing all four operations transforms, all the coherent states become $S-frac{1}{2}$ states, thus their inner product is one. Thus we obtain the second equality:
The third equality can be obtained using the definitions of the Schwinger boson coordinates in terms of the spherical coordinates
$$u = \cos\frac{\theta}{2}e^{i\frac{\phi}{2}}$$
$$v = \sin\frac{\theta}{2}e^{-i\frac{\phi}{2}}$$
The term $ u_i^{*}u_j+ v_i^{*}v_j$:
When one of the points is at the north pole ($u_i=1, v_i=0$) this expression is just $ \cos\frac{\theta_j}{2}$. In this case, the scalar product of the two coherent vectors is
$$\Omega_i  \cdot \Omega_j = \cos\theta_j = 2 (\cos\frac{\theta_j}{2})^2-1$$.
Since we can always place one of the points on the north pole, we have:
$$ u_i^{*}u_j+ v_i^{*}v_j = \sqrt{\frac{1 + \Omega_i  \cdot \Omega_j }{2}}$$
Up to a phase. This phase was computed by direct substitution of the coherent vectors by Auerbach equation (7.19), giving exactly the second equality in the question.
