Supercell construction and non-Abelian Bloch states in hyperbolic lattices

Hyperbolic {p,q}-lattices are tessellations of the sheet of constant negative curvature such that q copies of regular p-sided polygons meet at each vertex. The spectra of such lattices were recently characterized with hyperbolic band theory. Here, given a concrete hyperbolic lattice, one first constructs the hyperbolic translation group (HTG). The fundamental domain of the HTG defines a hyperbolic unit cell, and its Abelian representations span a four- (or higher-) dimensional Brillouin zone of Abelian Bloch states. However, this approach fails to capture non-Abelian Bloch states which transform in higher-dimensional representations.

In our work, we obtain novel insights into non-Abelian Bloch states by studying hyperbolic supercells. These correspond to fundamental domains of a reduced hyperbolic translation group, constructed according to a precise mathematical recipe. Curiously, instead of Brillouin zone folding that is observed in Euclidean supercells, hyperbolic supercells result in Brillouin zones with enlarged dimensions. We investigate the convergence of spectra as the supercell size grows and apply the technique to simple models of hyperbolic topological insulators.