Topological non-Abelian Gauge Structures in Cayley-Schreier Lattices

Recently, novel crystalline constructions known as Cayley-Schreier lattices have been suggested as a platform for realizing arbitrary gauge fields in synthetic crystals with real hopping amplitudes. We show that Cayley-Schreier lattices can naturally give rise to implementable lattice systems that incorporate non-Abelian gauge structures transforming into a space-group symmetry. We show that the symmetry sectors can, moreover, be interpreted as blocks of spin models that can effectively realize a wealth of different topological invariants in a single setup. We underpin these general results with concrete models and show how they can be implemented in current experimental platforms. Our work sets the stage for a systematic investigation of topological insulators and metals with non-Abelian gauge structures.