a prior Jordan-Wigner transformation, transforming the spin operators to proper fermionic operators, is not applicable in this case, since the interaction part is linear in the spin operators, so the Jordan-Wigner factors do not cancel out. Thus both transformations cannot be carried out exactly without increasing the descriptional complexity of the problem. Nevertheless the Hamiltonian can be diagonalized by a Fourier-transform in an approximate way.\
As said above, all modes decouple at $t_d=0$. For this reason, the spin-operators are in this limit equivalent to fermionic operators. If we assume that this replacement also holds for small values of $t_d$, the JCH model (\[eq:JCHreal\]) can be rewritten in a fermionic approximation $$\begin{gathered}
  \hat H = \omega \sum_{ j} \hat a_{ j}^\dagger \hat a_{ j} + \Delta \sum_{ j} 
\hat c^\dagger_{ j} \hat c_{ j} + g\sum_{ j} \left(\hat c ^\dagger_{ j} \hat a_{ j} + \hat a_{ j}^\dagger \hat c_{ j}\right)\\
   +\sum_{d} t_{ d} \sum_{ j} \left(\hat a^\dagger_{ j} \hat a_{ j+ d} + \hat a^\dagger_{ j+ d} \hat 
a_{ j}\right).\end{gathered}$$ Here the spin operators $\hat\sigma^+$ ($\hat \sigma^-$) are replaced by fermionic operators $\hat c^\dagger$ ($\hat c$). Within this approximation, a Fourier transform of both, the bosonic and fermionic degrees of 