since a larger dimension will affect calculations only by changing the Brillouin zone.

Let us concentrate on the case in which each lattice site is coupled to some external bath through a pumping mechanism as in Eq. . The master equation  during the annealing protocol can be easily integrated via a straightforward generalisation of the time-dependent Bogoliubov method already employed by Dziarmaga [@Dziarmaga_2005], as detailed in App. \[app:TraslInv\_Decay\]. The crucial point resides in the fact that, as for the Hamiltonian, the dissipative part of the Lindbladian with $L^{(1)}_n=c^{\dagger}_n$ does not mix the various modes at different momenta, once a Fourier transform has been employed. As a consequence, the density matrix at time $t$ factorizes into different contributions for the various modes: $$\rho(t) = \bigotimes_k \rho_k(t).
  \label{eq:RhoT_k}$$ The relevant Hilbert space for each positive momentum $k$ has dimension 4, and thus the Liouvillian dynamics can be easily followed inside it. We recall that, for the unitary Schrödinger dynamics, a further decomposition into independent $2 \times 2$ problems was possible, due to the additional conservation of the fermionic parity (which is now violated by the dissipative decaying terms).

![Final excess energy as a function of the annealing time, for the free-fermion model  