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 C₄T -- 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 8 corner modes at 0 or eigenphase, while the second type, hidden behind a weak topological phase, yields a unique phase with 8 corner modes at /2 eigenphase (after gapping out the counterpropagating edge states), arising from the product of particle-hole and phase rotation symmetry. By using a bulk Z₄ 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 Z₂ 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/n boundary modes, as well as suggests that such phases may find their experimental realization in coupled-ring-resonator networks.