Engineering and understanding of Cayley (and hyperbolic) crystals with prescribed nonabelian dynamics

Recently, Hamiltonian dynamics over hyperbolic lattices has gained attention not only for its novel theoretical aspects as alternatives of standard Euclidean lattices but also for its potential applications in cQED, quantum information theory, topological protection of states and critical phenomena. Mathematically, it has been realized that such systems, together with the Euclidean ones, are just an instance of a much broader class dubbed Cayley crystals, whose lattice sites are elements of a group while the Hamiltonian hoppings determine a Cayley graph on it. For physical applications, however, it is very profitable to be able to engineer crystals starting from desired local properties so as to have an understanding on the transport of states and the scalability of observables. In this talk I will show how to identify the "nonabelian skeleton" of a crystal and, conversely, how to build Cayley crystals starting from a given scalable structure. When the latter is finite, the Hamiltonian dynamics on Cayley crystals, comprising both abelian and nonabelian eigenstates, can be fully understood and is essentially equivalent to that of Euclidean crystals, but with a more structured unit cell. Moreover, such engineering of crystals allows to characterize explicitly all possible periodic boundary conditions in a straightforward way and suggests a way to embed them in real space so that, experimentally, it will be much easier to realize them than, e.g. in the hyperbolic case, using Poincarè-disk-like embeddings.