Higher-Order Topology and Fully Flat Bands in Multimode Photonic Time-Reversal-Invariant Lattices

Higher-order topological insulators (HOTIs) are unusual phases of matter hosting robust boundary-localized states of at least two dimensions less than the effective dimensionality of the sample. Their photonic realizations span a remarkably large spectrum of distinct HOTI phases, owing to a high versatility of such platforms. Furthermore, even more exotic phases could be engineered if the system additionally hosts flat bands in the spectrum. The latter may lead to stronger simultaneous localization of corner and bulk states, paving the way towards enhanced control of light propagation in photonic lattices. However, to date most designs involve either geometrically distorted photonic lattices or external effective magnetic flux constructed artificially by using additional off-resonant elements. In this work, we propose a different approach utilizing the interference of accidentally degenerate modes of the meta-atoms, which may lead both to unconventional higher-order topology and to nearly flat bands for geometrically undistorted time-reversal-invariant photonic lattices. We illustrate our proposal by characterizing three such models and outlining their similarities and unique features. Our approach could be harnessed to construct simple photonic designs with topologically and symmetry- protected control over the light propagation.