Topological edge and corner states in the biphenylene network

The electronic states and topological properties of the biphenylene network (BPN) are analyzed using a tight-binding model based on the $\ensuremath{\pi}$-electron network. It is shown that tuning the hopping parameters induces topological phase transitions, leading to the emergence of edge states owing to the nontrivial topological Zak phase of the bulk BPN. Elementary band analysis clearly gives the number of edge states, which are associated with the location of Wannier centers. In addition, we present the conditions for the emergence of corner states owing to the higher-order topological nature of the BPN.