shows both analytic approximations along with numerical data from DMRG [@Rossini2007] and mean field [@Greentree2006] calculations, where the modified hopping amplitude in the mean field Hamiltonian (\[eq:JCHMeanField\]) evaluates as $\widetilde J = t$. From the figure, it can be seen that the effective model gives a much better agreement with the numerical DMRG data, especially the slopes of the lobes agree perfectly at small hopping. The fermion approximation overestimates the size of the Mott lobe. In particular while the lower boundaries are rather well reproduced the upper boundaries have the wrong slope. Surprisingly though the critical hopping amplitudes seem to agree better with the DMRG data than the results obtained from the effective strong coupling Hamiltonians. Although the fermion approximation is quantitatively worse than the effective strong-coupling Hamiltonians, it provides a simple approximative solution to the JCHM beyond the mean field level which has the advantage of giving a closed form of the ground state.

Linear ion chain
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As a second example we consider a linear string of ions in an ion trap [@Ivanov2009], where the ions are coupled to an external laser field and interact with each other due to Coulomb repulsion via phonon exchange. This system is 