dimensional, consisting of the single state $\left|- , n\right \rangle$. Thus, up to first order, the energy is given by $E(nL)  =L E^-_n$. When adding an excitation, the local Hilbert space increases; now, (locally), the two states $\left|- , n\right \rangle$ and $\left|- , n+1\right \rangle$ need to be taken into account. So, in this limit, the system for an additional particle can be understood as a system consisting of effective spin $\frac 12$ particles. We will identify the states $\left| \Uparrow\right\rangle$ with the state $\left|- , n+1\right \rangle$ and $\left| \Downarrow\right\rangle$ with $\left|- , n\right \rangle$. In order to derive the effective spin $\frac12$ model, one has to look on the action of the hopping operator $\hat a_{j+1}^\dagger \hat a_j$ on the states in the Hilbert space. Using equations (\[eq:factorsA\]) to (\[eq:factorsD\]) and neglecting the contributions from the states $\left|+ , n\right \rangle$ and $\left|+ , n+1\right \rangle$, the hopping operator $\hat a^\dagger_{j+1} \hat a_j$ acts as $$\hat a^\dagger_{j+1} \hat a_j\left|\Downarrow\right \rangle_{j+1}\left|\Uparrow\right \rangle_j = B^-_n 
D^-_{n+1}\left|\Uparrow\right \rangle_{j+1}\left|\Downarrow\right \rangle_j$$ within the considered subspace. Therefore, by introducing spin operators ${\widetilde\sigma}^\pm_j$ the hopping term is equivalent to a nearest neighbor spin-spin interaction with $$\hat a^\dagger_{j+1} \hat a_j = B^-_n D^-_{n+1} {\widetilde\sigma}^+_{j+1} {\widetilde\sigma}^-_{j} 