critical hopping amplitude $t_{\rm crit}$ where $ \mu^+_n= \mu^-_n$, which is given by $$t_{\rm crit}/g = 2\frac{2 \sqrt n - \sqrt{n+1}-\sqrt{n-1}}{(\sqrt{n}+\sqrt{n+1})^2+(\sqrt{n+\delta_{n1}}+\sqrt{n-1})^2}.$$

Secondly we apply the second approximation to this model. With the given system parameters, the momentum dependent phonon energies (\[eq:PhononEnergies\]) are given by $$\begin{gathered}
\omega_k = \omega - 2  t \cos(2\pi\frac{   k  }L)\end{gathered}$$ and the energy in the $k$th momentum mode for a given filling $n$ reads (see eq. (\[eq:JCEnergyMomentum\])) $$\begin{gathered}
 E^{n}_{   k} = (1-\delta_{n0})\Biggl[n\omega-2 n\,  t \cos(2\pi\frac{   k  }L) + t \cos(2\pi\frac{   k  }L)\\
    -\sqrt{ t^2 \cos^2(2\pi\frac{   k  }L)+ n g^2}\Biggr]. \end{gathered}$$ Finally, following eqns. (\[eq:ChemicalPlusCommutator\]) and (\[eq:ChemicalMinusCommutator\]), the momentum modes $k^\prime$ ($k$) which minimize (maximize) the chemical potentials need to be found. In the present case ($t_1<0$), these are $k^\prime=0$ and $k=\frac L2$. Thus the resulting chemical potentials are $$\begin{gathered}
\mu^+_n-\omega= - 2 t + t \delta _{n0}-\sqrt{ t^2 +(n+1) g^2}\\
   +(1-\delta   _{n0}) \sqrt{   t^2+n g^2},\end{gathered}$$ for any $n$ and $$\begin{gathered}
\mu^-_n-\omega=   2 t -t\delta _{n1} - \sqrt{t^2+n g^2}\\
    +(1-\delta   _{n1}) \sqrt{t^2+ (n-1) g^2},\end{gathered}$$ for $n>0$. A closed form for the critical hopping can be found, but is rather lengthy and will therefore be skipped.\
We now compare our analytic results to various numerical calculations. Figure \[fig:DMRG\] 