B_{n,n+1} = - B_{n+1,n} = - J/2$. To enforce periodic boundary conditions, the following matrix elements are non-zero as well: $A_{1,L} = A_{L,1} = (-1)^{N_{\rm F}} J/2$ and $B_{L,1} = - B_{1,L} = (-1)^{N_{\rm F}} J/2$, where $(-1)^{N_{\rm F}}$ denotes the parity of the number of fermions $N_{\rm F} = \sum_{n} c^{\dagger}_n c_n$, which commutes with $H$. The Hamiltonian  can be exactly diagonalized using a Bogolibuov transformation [@LSM_1961; @Young_1997].

In the following we will completely relax the requirement on fermion-parity dependent boundary conditions, and we simply assume that anti-periodic boundary conditions for the fermions are always enforced. This assumption is perfectly justified for a purely coherent evolution, where the fermion parity is conserved and the initial ground state has an even number of fermions. The reason for enforcing the requirement even when considering the open-system adiabatic dynamics is that, as explained below, our Lindblad operators change the fermion parity, and it would be impossible to solve the problem by using a parity-dependent boundary conditions. More in detail, we will study below the adiabatic dynamics of $H(t)$ under the action of three different types of memoryless [*local*]{} environments. We model them in such a way that the Lindbladian  is a sum 