We consider electron transport in a nearly half-metallic ferromagnet, in which the minority spin electrons close to the band edge at the Fermi energy are Anderson-localized due to disorder. For the case of spin-flip scattering of the conduction electrons due to the absorption and emission of magnons, the Boltzmann equation is exactly soluble to the linear order. From this solution we calculate the temperature dependence of the resistivity due to single magnon processes at sufficiently low temperature, namelywhere L is the Anderson localization length and D is the magnon stiffness.And depending on the details of the minority spin density of states at the Fermi level, we find a T 1.5 or T 2 scaling behavior for resistivity. Relevance to the doped perovskite manganite systems is discussed.

We propose a general variational fermionic many-body wavefunction that generates an effective Hamiltonian in quadratic form which can then be exactly solved. The theory can be constructed within density functional theory framework and a self-consistent scheme is proposed for solving the exact density functional theory. We apply the theory to structurally disordered system and an symmetric and asymmetric Hubbard dimer and corresponding lattice models and the the single fermion excitation spectra show a persistent gap due to the fermionic entanglement induced pairing condensate. For disordered system, density of state at the edge of the gap diverges in the thermodynamic limit, suggesting a topologically ordered phase and a sharp resonance is predicted as the gap is not dependent on the temperature of the system. For the symmetric Hubbard model, the gap for both half filling and doped case suggests quantum phase transition between the AFM and SC is a continuous phase transition.

We apply the Self-Consistent Effective Hamiltonian Theory (SCEHT), which uses a general variational Fermionic many-body wave function to generate an effective Hamiltonian in a quadratic form, to the Anderson impurity model. The chiral symmetry-breaking quadratic effective Hamiltonian is solved exactly for the single Fermion excitation spectrum. We validate the theory by numerically solving a model problem. The solution shows the correct Kondo resonance in the quasi-particle density of states.

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Majorana fermion solution is obtained from the self-consistent effective Hamiltonian theory. The ground state is conjectured to be a non-empty vacuum with 2 fermions, one for each type. The first type is the original charged fermion and the second type the chiral charge-less Majorana fermion. The Marjorana fermion is like a shadow of the first fermion cast by the non-empty vacuum.