  This work presents the first lattice-QCD calculation of the twist-3 axial
quark GPDs for the proton using the large-momentum effective theory approach.
We calculate matrix elements with momentum-boosted proton states and a
non-local axial operator. The calculation is performed using one ensemble of
two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally
twisted mass fermions with a clover term. The ensemble has a volume
$32^3\times64$, lattice spacing 0.0934 fm, and corresponds to a pion mass of
260 MeV. The matrix elements are calculated for three values of the proton
momentum, namely 0.83, 1.25, and 1.67 GeV. The light-cone GPDs are defined in
the symmetric frame, which we implement here with a (negative) 4-momentum
transfer squared of 0.69, 1.38, and 2.76 GeV$^2$, all at zero skewness. We also
conduct several consistency checks, including assessing the local limit of the
twist-3 GPDs and examining the Burkhardt-Cottingham-type as well as
Efremov-Teryaev-Leader-type sum rules.
