Holstein Primakoff representation

[LagrangeEuler]

Holstein Primakoff representation in textbooks is defined by:
$$\hat{S}^+_m=\sqrt{2S}\sqrt{1-\frac{\hat{B}_m^+\hat{B}_m}{2S}}\hat{B}_m$$
$$\hat{S}_m^-=(\hat{S}^+_m)^+$$
$$\hat{S}_m^z=S-\hat{B}_m^+\hat{B}_m$$
And in practical cases it is often to use binomial series for square root, and condition for that is
$\frac{ \langle \hat{B}_m^+\hat{B}_m \rangle}{2S}<<1$ and because of that I can use binomial series only in case $S>>\frac{1}{2}$. However, it is very often to use that in case $S=\frac{1}{2}$ in papers. Any explanation?

 

[DrDu]

The Taylor series for the root still converges well for S=1/2 and $b^+b=1$ or 0 (other values don't occur). Test it yourself.