, \label{eq:effectiveEnergyMinus}
 \end{gathered}$$ where the same condition holds for $k^{\prime\prime}$. Now, putting the calculated energies (\[eq:effectiveEnergyPlus\]) and (\[eq:effectiveEnergyMinus\]) together, the chemical potentials and therefore the boundaries of the $n$th Mott insulating lobe can easily be derived. They are given by $$\begin{aligned}
  \mu^+_n&= E^-_{n+1}  - E^-_{n}  +   2 B^-_n D^-_{n+1}  \sum_d t_d \cos(2\pi\frac{   k^\prime    d}L)\label{eq:chemicalPotentialfirstPlus}\\
  \mu^-_n &=E^-_{n}   - E^-_{n-1}  -   2  B^-_{n-1} D^-_{n} \sum_d t_d \cos(2\pi\frac{   k^{\prime\prime}    d}L)\label{eq:chemicalPotentialfirstMinus} .
 \end{aligned}$$ where $k^\prime$ ($k^{\prime\prime}$) is chosen such that $\mu_n^+(k^\prime)$ ($\mu_n^-(k^{\prime\prime})$) is minimal (maximal). This result generalizes the findings from [@Schmidt2009; @Koch2009] to arbitrary hoppings $t_d$.

Fermion approximation {#sec:commapprox}
---------------------

In this subsection, we will apply an even simpler, but not that obvious approximation. When looking at the JCH Hamiltonian (\[eq:JCHreal\]), it can be seen that all terms are quadratic. These kinds of models are in general suited for an exact solution by means of a Fourier transform. The problem at this point is however, that the commutation relations of spin operator $\hat\sigma_j^\pm$ are not as simple as that of bosons or fermions. This limits the applicability of a Fourier transform, since the operators in momentum space will not obey the same commutation relation as in real space [^1]. The usual step of 