The SU(2) generalization of the U(1) slave-boson has been introduced in PRL 76, 503 (1996), and PRB 86, 085145 (2012). (A generic recipe for constructing the SU(2) slave-particle framework has been discussed Here.) In this framework, the electronic annihilation operator with spin $\sigma$ can be expressed as
$$
c_{\sigma}=\frac{1}{\sqrt{2}} (b^{\dagger}_{1} f_{\sigma} -b^{\dagger}_{2}f^{\dagger}_{\overline{\sigma}}),
$$
where $b~(b^{\dagger})$ is a slave-boson annihilation (creation) operator, and $f^{\dagger}_{i}(f_{i})$ creates (annihilates) a fermion with spin $i$. (The Nambu-like version of the above expression is presented in this link.)
In the SU(2) slave-boson language, it is claimed that "the double occupied state is automatically ruled out".
As a result, the one-site electronic states will be translated into the SU(2) slave-boson language as
$$
\begin{align}
|0 \rangle_{c} &= \frac{1}{\sqrt{2}}(b^{\dagger}_{1} + b^{\dagger}_{2} f^{\dagger}_{\uparrow} f^{\dagger}_{\downarrow}) |0\rangle_{sb},\\
c^{\dagger}_{\uparrow}|0 \rangle_{c} &= f^{\dagger}_{\uparrow} |0\rangle_{sb},\\
c^{\dagger}_{\downarrow}|0 \rangle_{c} &= f^{\dagger}_{\downarrow} |0\rangle_{sb},\\
c^{\dagger}_{\uparrow}c^{\dagger}_{\downarrow}|0 \rangle_{c} &=
\frac{1}{\sqrt{2}} ( b_{1} f^{\dagger}_{\uparrow}f^{\dagger}_{\downarrow} -b_{2}) |0\rangle_{sb}=0
\end{align}
$$
This result is a consequence of two points:

- The SU(2) nature of this representation which is under the constraint of
$$
b^{\dagger}_{1}b_{1} -b^{\dagger}_{2}b_{2}+\sum\limits_{\sigma \in \{\uparrow, \downarrow\}} f^{\dagger}_{\sigma}f_{\sigma}=1.
$$
- Exploiting, at most, only one species of each auxiliary particles per state. In other words, enforcing
$$
\forall \alpha \in \{1,2\} \rightarrow (b^{\dagger}_{\alpha}b_{\alpha} )^{2} =b^{\dagger}_{\alpha}b_{\alpha},
\qquad
\forall \sigma \in \{\uparrow,\downarrow\} \rightarrow
(f^{\dagger}_{\sigma}f_{\sigma} )^{2} =f^{\dagger}_{\sigma}f_{\sigma}.
$$

The question is that what is the necessity of applying the second implicit constraint? In addition, why we can not define a background charge for the vacuum of slave-boson such that the double-occupied state can survive?

This post imported from StackExchange Physics at 2017-07-02 10:53 (UTC), posted by SE-user Shasa