The non-Hermitian Aharonov-Bohm (AB) cage is a unique localization phenomenon that confines all possible excitations. This confinement leads to fully flat spectra in momentum space, which are typically accompanied with the degeneracy with various types. Classifying the degeneracy type is crucial for studying the dynamical properties of the non-Hermitian AB cage, but the methods for such classification and their physical connections are not very clear. Here, we construct a non-Hermitian AB cage in a bosonic Bogoliubov-de Gennes system with various types of degenerate flat bands (DFBs). Using the transfer matrix, we demonstrate the localization mechanism for the formation of an AB cage and derive the minimal polynomial in mathematics for classifying the degeneracy types of DFBs, thus providing comprehensive understanding of the correspondence among the degeneracy type of DFBs, the minimal polynomial, and the transfer matrix. With such correspondence, we propose a scheme to realize highly degenerate flat bands.

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• Aharonov-Bohm effect
• Localization
• Non-Hermitian systems
• Non-Abelian models