Band sorting based on global continuity of eigenvalues and topological properties of phononic crystals

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Band sorting is critical to obtaining physical properties from eigenvalues and eigenvectors that constitute the band diagram. We propose a band sorting method based on the global continuity and smoothness of the eigenvalues on the parameter space. Several strategies based on the connection between neighbor eigenvalues and how to sweep the parameter space are introduced to recognize level crossing degeneracies and level repulsion degeneracies. Eigenvectors are then reassigned to each band according to their corresponding eigenvalues. As an example, the band diagram of a 2D phononic crystal is sorted, and the group velocity and Berry curvature are compared before and after band sorting. The Berry curvature shows unexpected eigenvector discontinuities, indicating that our method based on the eigenvalues only is more reliable than methods based on the eigenvectors. Our proposed method is general and allows for studying the transport and topological properties of periodic materials.