Chern numbers in quantum graphs

Quantum graphs provide an analytically tractable setting for the study of Chern numbers and band degeneracies in periodic systems. We study the Chern numbers of energy bands in a two-dimensional square lattice quantum graph. We approach the problem by mapping the lattice to a single-vertex quantum graph with two loops of equal lengths pierced by magnetic fluxes. By establishing the degeneracy condition for its energy levels, we show that the model possesses two topological phases: a trivial phase, where the Chern numbers of all energy bands are $0$, and the nontrivial one, where the Chern numbers of successive energy bands alternate between $\pm1$. By applying the degeneracy condition, we calculate Chern-number phase diagrams analytically as a function of the node scattering matrix parameters and compare the results with numerical calculations.