  Detailed theoretical derivation is given for the tunnel magnetoresistance via
phonon-assisted hopping through an impurity chain under small magnetic field
and a large bias window. This derivation provides a rigorous basis for the
physical picture of Pauli blockade switch proposed in our previous paper
(arXiv:1404.0633). This picture captures the competition of external magnetic
field and internal spin interactions in the tunnel barrier, and relies
critically on the strong on-site Coulomb correlation at the impurities. The
master equations are obtained by deriving the equations of motion for the Green
functions at the impurity sites in a slave-boson representation, and utilizing
the so-called Langreth theorem to finally express the spin-dependent density
matrix in terms of the equilibrium distributions of the contact electrons and
of the phonon reservoir.
