Liouvillian and not necessarily to many-body characteristics.

The paper is organised as follows. In Sec. \[sec:Model\] we define the Hamiltonian models, the various dissipation schemes, and the annealing problem under investigation. We then study the departure from the instantaneous ground state, in the presence of dissipative processes which may cause decay or dephasing, both for a translationally invariant free-fermion model \[sec:TraslInv\], and for an Ising spin chain (Sec. \[sec:Ising\]). We end with a discussion of our findings and with the concluding remarks in Sec. \[sec:Conclusions\]. Technical details on the calculations for the quadratic fermionic model are provided in the Appendix.

Adiabatic dynamics with local dissipation: From Ising systems to fermion chains {#sec:Model}
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One of the simplest (and exactly solvable) models exhibiting a quantum phase transition is the one-dimensional Ising chain [@Sachdev]. This is defined by the Hamiltonian $$\label{eq:Ham_Ising}
  H(t) = - J \sum_{n} \sigma^{x}_{n}\sigma^{x}_{n+1} - \Gamma(t) \sum_{n} \sigma^{z}_{n},$$ where $\sigma^{\alpha}_{n} (\alpha = x,y,z)$ are the spin-$1/2$ Pauli operators for the $n$-th spin of the chain, while $J$ and $\Gamma(t)$ are respectively the coupling strength between neighbouring spins and the transverse magnetic field. We assume periodic boundary conditions, in such a way as to preserve translational invariance of the model. 