Correspondence of topological classification between quantum graph extra dimension and topological matter

In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there is a nontrivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of noninteracting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in noninteracting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.

Automated summary

This paper investigates the properties of five-dimensional Dirac fermions on quantum graphs, providing a complete topological classification of boundary conditions in terms of noninteracting fermionic topological phases. The study identifies ten symmetry classes of topological phases and gives topological numbers of the boundary conditions, predicting the presence of four-dimensional massless fermions localized at the vertex of the quantum graphs, which govern the low energy physics in four dimensions.