an excitation gives the maximum (minimum) reduction (increase) in the total energy. This yields $$\begin{aligned}
  \mu_n^+ &=  E_{k^\prime}^{n+1}-E_{k^\prime}^{n}\label{eq:ChemicalPlusCommutator}\\
  \mu_n^- &=  E_{k}^{n}-E_{k}^{n-1},\label{eq:ChemicalMinusCommutator}\end{aligned}$$ where $k^\prime$ ($k$) is chosen such that $\mu_n^+(k^\prime)$ ($\mu_n^-(k)$) is minimal (maximal). The actual values of $k$ and $k^\prime$ depend mainly on the sign of the hopping amplitudes $t_d$.

Application to specific realizations of the JCH model {#sec:Application}
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After having introduced the two approaches used in this paper, we will apply them to the case of the simple JCHM with positive effective mass and nearest neighbor hopping and to a modified model describing the physics of a linear ion chain. The case of the simple JCHM essentially serves as a testing ground for our approximation schemes including a comparison of the analytic results to numerical data from DMRG and mean field calculations. Later on, the generalized JCHM will be treated by both approximations giving analytic results for the phase diagram in a wide range of parameters.

JCHM with positive effective mass and nearest-neighbor hopping
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Without loss of generality, we will specialize here on the case discussed in [@Rossini2007]. The Hamiltonian of the JCHM in this case is given by $$\begin{gathered}
 \hat H = \omega \sum_{ j} \hat 