Topological junction states and their crystalline network in systems with chiral symmetry: Application to graphene nanoribbons

We develop a general theoretical framework based on Z classification to count the number of topological bound states at a junction of chiral-symmetric one-dimensional systems. The formulation applies to general multiway junctions composed of an arbitrary number of channels and an arbitrary joint structure. By using the formula, we calculate the zero-energy bound states in various types of two-way and three-way junctions of semiconducting graphene nanoribbons. We then consider periodic two-dimensional networks of graphene nanoribbons and show that the topological junction states form isolated energy bands inside the bulk energy gap, which can be viewed as a two-dimensional crystal of the effective atoms. Depending on the Z number of a single junction, we have a different set of effective atomic orbitals, resulting in various types of nanoscale metamaterials, which are often accompanied by flat bands. The system would provide an ideal platform for quantum simulator to emulate a strongly-interacting fermion system on various types of lattices.