Characterizing topology beyond perfect crystals

Topological phases are often formulated through Hamiltonians in momentum space, but topology is ultimately a property of the quantum state itself. I present a one-particle density matrix (OPDM) framework for characterizing topology directly from the state, applicable to disordered, interacting, and higher-order systems. In this approach, the one-particle density matrix encodes all topological information and remains valid without translation invariance. We derive the chiral and Chern–Simons markers, real space topological invariants in odd dimensions in terms of the OPDM. When the OPDM spectrum is gapped, it can be adiabatically flattened to a topologically equivalent projector, extending these markers to interacting states.Finally, we reformulate the mode–shell correspondence in terms of the OPDM framework, linking bulk markers to boundary modes and applying it to amorphous higher-order topological insulators. This provides a state-based characterization of topology beyond crystalline and noninteracting limits.