Network model for magnetic higher-order topological phases

We propose a network-model realization of magnetic higher-order topological phases (HOTPs) in the presence of the combined space-time symmetry C 4 T —the product of a fourfold rotation and time-reversal symmetry. We show that the system possesses two types of HOTPs. The first type, analogous to Floquet topology, generates a total of eight corner modes at 0 or π eigenphase, while the second type, hidden behind a weak topological phase, yields a unique phase with eight corner modes at ± π / 2 eigenphase (after gapping out the counterpropagating edge states), arising from the product of particle-hole and phase-rotation symmetry. By using a bulk Z 4 topological index ( Q ) , we found both HOTPs have Q = 2 , whereas Q = 0 for the trivial and the conventional weak topological phase. Together with a Z 2 topological index associated with the reflection matrix, we are able to fully distinguish all phases. Our work motivates further studies on magnetic topological phases and symmetry-protected 2 π / n boundary modes, as well as suggesting that such phases may find their experimental realization in coupled-ring-resonator networks.