  Slave-particle method is a powerful tool to tackle the correlation effect in
quantum many-body physics. Although it has been successfully used to comprehend
various intriguing problems, such as Mott metal-insulator transition and Kondo
effect, there is still no convincing theory so far on the availability and
limitation of this method. The abuse of slave-particle method may lead to wrong
physics. As the simplest slave-particle method, $\mathbb{Z}_2$ slave spin,
which is widely applied to many strongly correlated problems, is highly
accessible and researchable. In this work, we will uncover the nature of
$\mathbb{Z}_2$ slave-spin method by studying a two-site Hubbard model. After
exploring some properties of this toy model, we make a comparative analysis of
the results obtained by three methods: (i) slave-spin method on mean-field
level, (ii) slave-spin method with gauge constraint and (iii) the exact
solution as a benchmark. We find that, protected by particle-hole symmetry, the
slave-spin mean-field method can recover the static properties of ground state
exactly at half filling. Furthermore, in the parameter space where both $U$ and
$T$ are small enough, slave-spin mean-field method is also reliable in
calculating dynamic and thermal dynamic properties. However, when $U$ or $T$ is
considerably large, the mean-field approximation gives ill-defined behavior,
which results from the unphysical states in enlarged Hilbert space. These
findings lead to our conclusion that the accuracy of slave particle can be
guaranteed if we can exclude all unphysical states by enforcing gauge
constraints.Our work demonstrates the promising prospect of slave-particle
method in studying complex strongly correlated models with specific symmetry or
in certain parameter space.
