of terms that act uniformly ($\kappa_n = \kappa, \; \forall n$) on each site $n$ of the chain: $$\begin{aligned}
  i)   \;\; & L_{n}^{\rm (1)} = c^{\dagger}_{n}         && \mbox{pumping mechanism} , \label{eq:L_pump}\\
  ii)  \;\; & L_{n}^{\rm (2)} = c_{n}                   && \mbox{decaying mechanism} , \label{eq:L_decay}\\
  iii) \;\; & L_{n}^{\rm (3)} = c^{\dagger}_{n} c_{n}   && \mbox{dephasing mechanism} . \label{eq:L_dephas}\end{aligned}$$ Note that while the fermionic dephasing environment  can be directly mapped into a dephasing environment of spins, since there is direct local mapping $c^{\dagger}_{n} c_{n} \to \tfrac12 (\sigma^{z}_{n}+1)$, the same is not true for the decay and pumping. Indeed, when mapping from a spin operator $\sigma^{-}_{n}$ ($\sigma^+_n$) to a fermionic operator $c_{n}$ ($c^\dagger_n$), the Jordan-Wigner transformation  includes a non-local operator (string). We should stress that the naming of these Lindblad operators has been chosen with respect to their action on a system with a local, diagonal Hamiltonian, and not with respect to their actual effect on the system that we study. The choice of these specific Lindblad operators is motivated by the possibility to study, in an essentially exact way, the competition between the unitary dynamics and dissipative effects, focusing on features that do not depend qualitatively on the form of the coupling to 