Hyperbolic lattices: Experimentally Realizable Cayley Crystal Decomposition

Hyperbolic lattices, characterized by negative curvature and non-commutative translations, offer a rich playground for exploring exotic electronic states. This talk explores breaking down hyperbolic lattices into curved Euclidean lattices (amenable to strain engineering in graphene) and simpler non-Abelian Z₂ lattices generated by non-commuting translations (Cayley crystals). I will present our very recent findings (still under progress), revealing two distinct classes of states within these Z₂ lattices: Abelian states exhibiting conventional behavior and non-Abelian states experiencing a surprising Hall drift motion under an electric field. This intriguing result suggests the presence of an effective internal magnetic field in the non-Abelian sector, opening exciting avenues for investigating novel physical phenomena.