representation of the chemical potentials are left out here. They can be found straight forwardly just as in the case of the simple cubic JCH model.\
When following the approximative method from section \[sec:commapprox\], the Hamiltonian for the uncoupled JC models is given by eq. (\[eq:JCHMomentum\]), with the phonon energies being $$\begin{gathered}
\omega_k = -\omega + 2 t \sum_{   d} \frac{ \cos(2\pi\frac{   k    d}L)}{d^3},\end{gathered}$$ according to (\[eq:PhononEnergies\]). Note that since $\omega = 2 t\, \zeta (3)$, all $\omega_k$’s are negative. Using the polylogarithm $\text{Li}_n\left(x\right)=\sum_{d=1}^\infty \frac{x^d}{d^n}$ one can write them in the explicit form $$\begin{aligned}
\omega_k= t\left[ \text{Li}_3\left(e^{2 \pi i \frac kL }\right)+\text{Li}_3\left(e^{-2 \pi i \frac kL }\right)-2\zeta(3)\right]. \label{eq:omegak}\end{aligned}$$ The minimum value of $\omega_k=-7 t \zeta(3)/2$ is attained for $k=\frac{L}{2}$ as expected from the positive sign of the hopping term. The energies for each momentum mode are given by the solution (\[eq:JCEnergyMomentum\]) of the JC model and the corresponding spectrum is shown in fig. \[fig:EnergiesFilling\].\

From the knowledge of the dispersion relation for different fillings, it is now easy to construct the phase diagram. As discussed in section \[sec:commapprox\], the flat dispersion for $t=0$ leads to the ground state having an equal number of excitations in every momentum mode $k$. The 