  The topological charge-Hall effect (TCHE) and the topological spin-Hall
effect (TSHE), arising from ferromagnetic (FM) and antiferromagnetic (AFM)
skyrmions, respectively; can be elucidated through the emergence of
spin-dependent Berry gauge fields that affect the adiabatic flow of electrons
within the skyrmion texture. TCHE is absent in systems with parity-time (PT)
symmetry, such as collinear AFM systems. In this paper, we theoretically study
TCHE and TSHE in a canted antiferromagnetic within the diffusive transport
regime. Spin canting or weak ferromagnetism in canted AFMs, which break the PT
symmetry, may arise, e.g., from strong homogeneous Dzyaloshinskii-Moriya
interactions. Using a semiclassical Boltzmann approach, we obtain diffusion
equations for the spin and charge accumulations in the presence of finite
spin-flip and spin-dependent momentum relaxation times. We show that the weak
ferromagnetic moment stemming from spin canting and the subsequent breaking of
parity-time symmetry results in the emergence of both finite TCHE and TSHE in
AFM systems.
