well described by a modified JCH model with a specific short range hopping with negative effective mass and site dependent parameters. First we will shortly introduce the model and give a derivation of the corresponding homogeneous limit. Afterwards we will apply the both approximations and discuss the phase boundaries within these approximation giving explicit analytic results for them.\
As shown in [@Ivanov2009], the Hamiltonian of a linear string of $L$ ions simultaneously irradiated by a laser which is tuned close to the red radial motional sideband and in the Lamb-Dicke regime is given by $$\begin{gathered}
  \hat H =  \sum_{j=0}^{L-1} \omega_j\hat a_j^\dagger \hat a_j + \Delta \sum_j \hat\sigma^\dagger_j 
\hat\sigma^-_j + g\sum_j
\left(\hat \sigma ^\dagger_j \hat a_j + \hat a_j^\dagger \hat\sigma_j^-\right) \\
 + \sum_{j} \sum_{d=1}^{L-j} t_{j,j+d} \left( \hat a^\dagger_{j+d} \hat a_{j} + \hat a^\dagger_{j} \hat 
a_{j+d}\right).\label{eq:JCHIonChain}\end{gathered}$$ Here $\hat a_j^\dagger$ and $\hat a_j$ describe the creation and annihilation of a local phonon at the $j$th site (ion), $\hat \sigma_j^\pm$ are the spin flip operators between the internal states of the ion, $\Delta$ is the detuning of the external laser field from the red motional sideband. $g$ describes the phonon-ion coupling in the Lamb Dicke limit; for a precise definition see [@Ivanov2009]. The local 