  We obtain the analytical expression for the Kraus decomposition of the
quantum map of an environment modeled by an arbitrary quadratic fermionic
Hamiltonian acting on one or two qubits, and derive simple functions to check
the non-positivity of the intermediate map. These functions correspond to two
different sufficient criteria for non-Markovianity. In the particular case of
an environment represented by the Ising Hamiltonian, we discuss the two sources
of non-Markovianity in the model, one due to the finite size of the lattice,
and another due to the kind of interactions.
