Mapping topological states with compacted dimensions

With conformal coordinate transformations, compacted dimensions might be realized in photonic systems. Here we show, with a one-dimensional (1D) photonic model system, that the essential topological nature of a two-dimensional (2D) nanoribbon is conserved in a mapping without conformal transformations. We take topological edge states as an example. They exist in physical systems of different dimensions, such as Su-Schrieffer-Heeger (SSH) model in 1D with dimerized polyacetylene, or a Haldane model in a 2D hexagonal lattice with spin-orbit coupling. In addition, the 2D graphene nanoribbon model also hosts topological edge states if the edge is of "zigzag" type with nontrivial Zak phase. In this Letter, we introduce a mapping formalism which establishes the correspondence between the momentum space of the nanoribbon and the parameter space of a 1D diatomic chain. It explicitly demonstrates that topological origin of the nanoribbon system can be reduced to the SSH model, which reveals the unification of topological nature across different dimensions. As an example, we reconstruct the 2D graphene nanoribbon band structure, including topological edge states, using 1D optical waveguide arrays. The coupling coefficients between waveguides are directly transformed from nanoribbons' electron momenta. We believe that it is a powerful tool to connect physical phenomena in different dimensions and offers a new experimental methodology based on photonic platforms for condensed-matter research.