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

TL;DR

This paper establishes a topological classification linking quantum graph boundary conditions with fermionic topological phases, revealing their role in predicting massless fermions in lower dimensions.

Contribution

It introduces a novel correspondence between quantum graph boundary conditions and topological phases, enabling a complete classification in five-dimensional fermionic systems.

Findings

Topological numbers classify boundary conditions.

Correspondence predicts localized massless fermions.

Complete classification of boundary conditions in topological terms.

Abstract

In this paper, we study five-dimensional Dirac fermions of which extra-dimension is compactified on quantum graphs. We find that there is a non-trivial 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 non-interacting 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 non-interacting 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.