  Studying many-body versions of Landau-Zener-like problems of non-interacting
electrons in the Slater formalism for several $k \cdot p$ models representing
Weyl and Dirac semimetals, we systematically include non-adiabatic corrections
to a quantum limit of chiral charge pumping in these models. In this paper, we
show that relative homotopy invariant [Sun et al., Phys. Rev. Lett. 121, 106402
(2018)] and Euler class invariant [Bouhon et al., Nat. Phys. 16, 1137 (2020)]
non-trivially manifest in the non-adiabatic corrections to the quantum limit of
chiral charge pumping. These corrections could affect conductivity channels
connected with the presence of chiral anomaly. Moreover, we show that, for
non-symmorphic systems, this contribution is sensitive to the direction of the
applied magnetic field (in respect to the so-called non-symmorphic nodal loop),
suggesting that the conjectured direction-selective chiral anomaly in
non-symmorphic systems [Bzdu\v{s}ek et al., Nature (London) 538, 75 (2016)]
could lead to a strongly anisotropic longitudinal magnetoresistance. The
presented approach can be easily applied to other $k \cdot p$ or tight-binding
models.
