than zero to yield useful results.

Approximative determination of the phase boundaries {#sec:Approximations}
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Effective strong-coupling model {#sec:effective}
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From the discussion above, it can be seen that the phase boundaries are defined by the closure of the particle-hole gap. In the present subsection, we will derive effective Hamiltonians in the strong-coupling limit for the calculation of the upper and lower chemical potential of the $n$th Mott lobe, allowing to calculate the particle-hole gap in first order of the hopping amplitudes $t_d$. To do so, we employ degenerate perturbation theory using Kato’s expansion as summarized in [@Klein1974] in first order with $H_{\rm eff} = \mathcal P V\mathcal P$. This procedure is equivalent to the polariton mapping considered in [@Angelakis2007; @Schmidt2009]. First, we note that according to eq. (\[eq:JCEnergy\]) the state $\left| + , n\right \rangle $ is separated by a large energy gap from the ground state $\left|- , n\right \rangle $. Thus, $\left| + , n\right \rangle $ can be completely neglected in the following as already mentioned in [@Angelakis2007; @Koch2009].\
When looking for the energy of the ground state with $N=nL$, from perturbation theory, no 1st order contributions are present. So, the Hilbert space per site is one 