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