is not that straight forward. When simply calculating the modified hopping amplitude $\widetilde J=-t\sum_d  \frac1{d^3} = - t \zeta(3)$, the hopping gets negative and therefore mean field theory inapplicable. This problem can be overcome by first applying a canonical transformation to all used operators. The transformation $$\hat a_j \mapsto (-1)^j \hat a_j$$ for the annihilation operator and accordingly to all the other operators $\hat a^\dagger_j$,$\hat\sigma_j^\pm$, maps the JCH model (\[eq:JCHreal\]) again to the JCH model, but with $t_d \mapsto (-1)^d t_d$. After this transformation, the modified hopping evaluates to $\widetilde J = - t\sum_d \frac{(-1)^d}{d^3}=3t \zeta (3)/4$ being positive. Now the application of the mean field theory is straight forward, following the usual route.

After having introduced the homogeneous limit of the model, the approximations introduced in section \[sec:Approximations\] will both be applied. Starting with the effective strong-coupling theory from section \[sec:effective\], the chemical potentials for the upper and lower boundary of the lobes are given by eqns. (\[eq:chemicalPotentialfirstPlus\]) and (\[eq:chemicalPotentialfirstMinus\]). The proper momentum modes $k^\prime,k^{\prime\prime}$, which minimize (maximize) the chemical potentials are both found to be $k=L/2$. This results from the negative mass. Due to the complexity of the problem, especially the analytic form of $B_n^-$ and $D_n^-$, analytic 