Gapless higher-order topology and corner states in Floquet systems

Higher-order topological phases (HOTPs) possess localized and symmetry-protected eigenmodes at corners and along hinges in two and three dimensional lattices. The numbers of these topological boundary modes will undergo quantized changes at the critical points between different HOTPs. In this work, we reveal unique higher-order topology induced by time-periodic driving at the critical points of topological phase transitions, which has no equilibrium counterparts and also goes beyond the description of gapped topological matter. Using an alternately coupled Creutz ladder and its Floquet-driven descendants as illustrative examples, we analytically characterize and numerically demonstrate the zero and $\pi$ corner modes that could emerge at the critical points between different Floquet HOTPs. Moreover, we propose a unified scheme of bulk-corner correspondence for both gapless and gapped Floquet HOTPs protected by chiral symmetry in two dimensions. Our work reveals the possibility of corner modes surviving topological transitions in Floquet systems and initializes the study of higher-order Floquet topology at quantum criticality.