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Higher-order Fermi-liquid corrections for an Anderson impurity away from half-filling III: non-equilibrium transport

We extend the microscopic Fermi-liquid theory for the Anderson impurity [Phys.\ Rev.\ B {\bf 64}, 153305 (2001)] to explore non-equilibrium transport at finite magnetic fields. Using the Ward identities in the Keldysh formalism with the analytic and anti-symmetric properties of the vertex function, the spin-dependent Fermi-liquid corrections of order $T^2$ and $(eV)^2$ are determined at low temperatures $T$ and low bias voltages $eV$. Away from half-filling, these corrections can be expressed in terms of the linear and non-linear static susceptibilities which represent the two-body and three-body fluctuations, respectively. We calculate the non-linear susceptibilities using the numerical renormalization group, to explore the differential conductance $dI/dV$ through a quantum dot. We find that the two-body fluctuations dominate the corrections in the Kondo regime at zero magnetic field. The contribution of the three-body fluctuations become significant far away from half-filling, especially in the valence-fluctuation regime and empty-orbital regimes. In finite magnetic fields, the three-body contributions become comparable to the two-body contributions, and play an essential role in the splitting of the zero-bias conductance peak occurring at a magnetic field of the order of the Kondo energy scale. We also apply our microscopic formulation to the magneto-resistance and thermal conductivity of dilute magnetic alloys away from half-filling.