oscillation frequencies $\omega_j$ and the hopping amplitudes $t_{j,j+d}$ are determined by the longitudinal and transversal trap frequencies $\omega_z$ and $\omega_x$ via $$\omega_j = -\frac{\omega_z^2}{2\omega_x} \sum_{\begin{matrix} \scriptstyle l=0\\\scriptstyle l\not = j \end{matrix}}^{L-1} \frac{1}{| u_j - u_l|^3}
 \hspace{0.5cm} 
t_{j, j+d} = \frac{\omega_z^2}{2\omega_x}  \frac{1}{| u_j - u_{j+d}|^3}, \label{eq:Constants}$$ where $u_j$ are the equilibrium positions of the ions [@James1998]. For sufficiently large $L$, the equilibrium positions of the ions at the center are approximately equidistant, giving $u_j = j\, \widetilde u$, with $\widetilde u$ being the distance of two adjacent ions.

Let us now discuss the limit of a homogeneous chain neglecting any boundary effect. In this limit eqs. (\[eq:Constants\]) can be rewritten for $L\to\infty$, yielding position independent phonon energies $\omega_j\equiv -\omega$ and hopping amplitudes $t_{j,j+d}\equiv t_d$ $$\begin{aligned}
{3}
t_d &= \frac{\omega_z^2}{2\omega_x \widetilde u^3} \frac 1 {d^3}&&=t\ \frac 1 {d^3} \label{eq:hopping},\\
 \omega&= 2\frac{\omega_z^2}{2\omega_x \widetilde u^3} \zeta(3)&&=2 t \  \zeta(3) \label{eq:omega},\end{aligned}$$ where $t = \frac{\omega_z^2}{2\omega_x \widetilde u^3}$ acts as a small parameter and $\omega>0$. $\zeta(x)$ is the Riemann $\zeta$-function.

One notices a negative oscillator energy $-\omega$ and a negative effective mass, which is a result of the positive hopping strength $t$. This negative mass is the reason, why the application of the mean field theory 