  Layered materials tend to exhibit intriguing crystalline symmetries and
topological characteristics based on their two dimensional (2D) geometries and
defects. We consider the diffusion dynamics of positively charged ions
(cations) localized in honeycomb lattices within layered materials when an
external electric field, non-trivial topologies, curvatures and cationic
vacancies are present. The unit (primitive) cell of the honeycomb lattice is
characterized by two generators, $J_1, J_2 \in \rm SL_2(\mathbb{Z})$ of modular
symmetries in the special linear group with integer entries, corresponding to
discrete re-scaling and rotations respectively. Moreover, applying a 2D
conformal metric in an idealized model, we can consistently treat cationic
vacancies as topological defects in an emergent manifold. The framework can be
utilized to elucidate the molecular dynamics of the cations in exemplar
honeycomb layered frameworks and the role of quantum geometry and topological
defects not only in the diffusion process such as prediction of conductance
peaks during cationic (de-)intercalation process, but also pseudo-spin and
pseudo-magnetic field degrees of freedom on the cationic honeycomb lattice
responsible for bilayers.
