  Using the non-equilibrium Green\noindent 's function method and the Keldysh
formalism, we study the effects of spin-orbit interactions and time-reversal
symmetry breaking exchange fields on non-equilibrium quantum transport in
graphene armchair nanoribbons. We identify signatures of the quantum spin Hall
(QSH) and the quantum anomalous Hall (QAH) phases in non-equilibrium edge
transport by calculating the spin-resolved real space charge density and local
currents at the nanoribbon edges. We find that the QSH phase, which is realized
in a system with intrinsic spin-orbit coupling, is characterized by chiral
counter-propagating local spin currents summing up to a net charge flow with
opposite spin polarization at the edges. In the QAH phase, emerging in the
presence of the Rashba spin-orbit coupling and a ferromagnetic exchange field,
two chiral edge channels with opposite spins propagate in the same direction at
each edge, generating an unpolarized charge current and a quantized Hall
conductance $G = 2 e^2/h$. Increasing the intrinsic spin-orbit coupling causes
a transition from the QAH to the QSH phase, evinced by characteristic changes
in the non-equilibrium edge transport. In contrast, an antiferromagnetic
exchange field can coexist with a QSH phase, but can never induce a QAH phase
due to a symmetry that combines time-reversal and sublattice translational
symmetry.
