Unified characterization for higher-order topological phase transitions

Higher-order topological phase transitions (HOTPTs) are associated with closing either the bulk energy gap (type-I) or boundary energy gap (type-II) without changing symmetry, and conventionally, both transitions are captured in real space and characterized separately. Here, we propose a momentum-space topological characterization of HOTPTs which unifies both types of topological transitions and enables a precise detection by quench dynamics. Our unified characterization is based on a correspondence between mass domain walls on real-space boundaries and higher-order band-inversion surfaces (BISs) which are characteristic interfaces in the momentum subspace. Topological transitions occur when momentum-space topological nodes, dubbed higher-order topological charges, cross the higher-order BISs after proper projection. Particularly, the bulk (boundary) gap closes when all (part of) topological charges cross the BISs, characterizing type-I (type-II) HOTPTs. These distinct dynamical behaviors of higher-order topological charges can be feasibly measured from quench dynamics driven with control in experiments. Our work opens an avenue to characterize and detect the two types of HOTPTs within a unified framework and shall advance research in both theory and experiments.