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While topology is a property of a quantum state itself, most existing methodsfor characterizing the topology of interacting phases of matter require directknowledge of the underlying Hamiltonian. We offer an alternative by utilizingthe one-particle density matrix formalism to extend the concept of the Chern,chiral, and Chern-Simons markers to include interactions. The one-particledensity matrix of a free-fermion state is a projector onto the occupied bands,defining a Brillouin zone bundle of the given topological class. This is nolonger the case in the interacting limit, but as long as the one-particledensity matrix is gapped, its spectrum can be adiabatically flattened,connecting it to a topologically equivalent projector. The correspondingtopological markers thus characterize the topology of the interacting phase.Importantly, the one-particle density matrix is defined in terms of a givenstate alone, making the local markers numerically favorable, and providing aninvaluable tool for characterizing topology of interacting systems when onlythe state itself is available. To demonstrate the practical use of the markerswe use the chiral marker to identify the topology of the ground state of theMajorana-XYZ model as well as the midspectrum eigenstates of the Ising-Majoranachain across the transition between the ergodic and many-body localized phases.

Interacting Local Topological Markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states