thus the excited states $ \left|+ , n\right \rangle$ do not contribute to the ground state.\
For the following discussion it will be useful to consider the action of a single bosonic creation or annihilation operator on a given JC eigenstate $\left| \pm , n\right \rangle$. Defining $$\begin{aligned}
 A^\pm_n&=\begin{cases} \sqrt n\ \alpha^\pm_n\beta^-_{n+1} \pm\sqrt{n+1}\ \beta_n^\pm\alpha^-_{n+1}
& n>0\\\hspace{2cm} \alpha^-_1 &n=0\end{cases}\label{eq:factorsA}\\
 B^\pm_n&=\begin{cases}\sqrt n \ \alpha^\pm_n\beta^+_{n+1} \mp\sqrt{n+1}\ \beta_n^\pm\alpha^+_{n+1}& n>0\\\hspace{2cm} 
-\alpha^+_1 &n=0\end{cases}\\
 C^\pm_n&=\begin{cases} \sqrt {n-1}\ \alpha^\pm_n\beta^-_{n-1} \pm\sqrt{n}\ \beta_n^\pm\alpha^-_{n-1}
& n>1\\ \hspace{2cm} 0&n\le1\end{cases}\\
 D^\pm_n&=\begin{cases} \sqrt {n-1}\  \alpha^\pm_n\beta^+   _{n-1} \mp\sqrt{n}\ \beta_n^\pm\alpha^+_{n-1}& n>1
\\\hspace{2cm} \pm \beta^\pm_1 \delta_{n,1}&n\le1\end{cases},\label{eq:factorsD}\end{aligned}$$ the action of $\hat a^\dagger $ and $\hat a$ on the state $\left| \pm , n\right \rangle$ can be seen to be $$\begin{aligned}
\hat a^\dagger  \left| \pm , n\right \rangle    &=  A^\pm_n\left| + , n+1\right\rangle  +  B^\pm_n  \left| - , n+1\right  \rangle\\
 \hat a \left| \pm , n\right \rangle            &=  C^\pm_n \left| + , n-1\right\rangle +  D^\pm_n\left| - ,
 n-1\right  \rangle   ,\end{aligned}$$ i. e. $\hat a^ \dagger$ and $\hat a$ connect the manifold of states $\left| \pm , n\right  \rangle$ to the manifolds $\left| \pm , n+1\right  \rangle$ and $\left| \pm , n-1\right  \rangle$ respectively as expected.\
In order to calculate the phase boundaries of the Mott insulating lobes for the 