Inherent altermagnetism in minimal tight-binding models of regular hyperbolic lattices
Altermagnets are a novel class of magnetic systems characterized by their momentum-dependent spin splitting without net magnetization. In this work, we extend established Euclidean tight-binding models of altermagnets to regular hyperbolic lattices in two spatial dimensions defined on a discretized Poincaré disk. Using hyperbolic crystallography and Abelian hyperbolic band theory, we show that the inclusion of next-nearest neighbor hopping is sufficient to induce spin splitting in minimal tight-binding models of bipartite hyperbolic lattices. While certain families and special cases of hyperbolic lattices remain antiferromagnetic, we identify an entire family and a special case that generically permit spin splitting in this framework. Hence, altermagnetism is inherent to certain hyperbolic lattices. Since Abelian hyperbolic band theory yields a momentum space that is at least four-dimensional, we classify the leading spin-splitting harmonics using four-dimensional atomic orbitals. As an outlook, we apply non-Abelian hyperbolic band theory to the $\{10,5\}$ lattice. Although its Abelian spectrum is spin degenerate, two selected higher-genus supercells exhibit spin splitting. This finding suggests, that spin degeneracy does not necessarily persist within all translation representation sectors.