a_{ j}^\dagger \hat a_{ j} + \Delta \sum_{ j} \hat\sigma^\dagger_{ j} \hat\sigma^-_{ j} + 
g\sum_{ j} \left(\hat\sigma ^\dagger_{ j} \hat a_{ j} + \hat a_{ j}^\dagger \hat\sigma_{ j}^-\right)\\
 -t \sum_{ j} \left(\hat a^\dagger_{ j} \hat a_{ j+ 1} + \hat a^\dagger_{ j+ 1} \hat 
a_{ j}\right).\label{eq:JCH}\end{gathered}$$ with $\omega=\Delta$. Comparing with the Hamiltonian eq. (\[eq:JCHreal\]), one notes that the hopping amplitudes satisfy $t_d = - t \delta_{d1}$.\
For the calculation of the chemical potentials, we first have to determine the momentum modes $k^\prime$ and $k^{\prime\prime}$ which contribute to the energy. For $\omega=\Delta$, the coefficients in (\[eq:JCGroundState\]) are $\alpha_n^\pm = \frac 1{\sqrt 2} = \beta_n^\pm$ and therefore $$B_n^- = \left \{ \begin{matrix}\frac{\sqrt n+\sqrt{n+1}}{2} &&\hspace{1cm} n>0\\ -\frac{1}{\sqrt 2} &&\hspace{1cm} n=0 
\end{matrix}\right\} = D_{n+1}^-.$$ With this, the function $F_n(k)$ is given by $$F_n(k) = - t \frac{\left(\sqrt n+\sqrt{n+1}\right)^2}{2-\delta_{n,0}} \cos(2\pi \frac kL).$$ The both chemical potentials have its minimum (maximum) at $k=0$. Putting everything together, the phase boundaries of the $n$th Mott lobe, calculated using the effective strong-coupling model read $$\begin{aligned}
 \mu^+_n &= \omega-\frac 12 \chi_{n+1} + \frac{1-\delta_{n0}}2 \chi_n -t\frac{\left(\sqrt 
n+\sqrt{n+1}\right)^2}{2-\delta_{n0}}, \\
\intertext{for any $n$ and}
 \mu^-_n &= \omega-\frac 12 \chi_{n} + \frac{1-\delta_{n1}}2 \chi_{n-1} +t\frac{\left(\sqrt 
n+\sqrt{n-1}\right)^2}{2-\delta_{n1}},\end{aligned}$$ for $n>0$. This allows for the determination of the 