  While topology is a property of a quantum state itself, most existing methods
for characterizing the topology of interacting phases of matter require direct
knowledge of the underlying Hamiltonian. We offer an alternative by utilizing
the one-particle density matrix formalism to extend the concept of the Chern,
chiral, and Chern-Simons markers to include interactions. The one-particle
density 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 no
longer the case in the interacting limit, but as long as the one-particle
density matrix is gapped, its spectrum can be adiabatically flattened,
connecting it to a topologically equivalent projector. The corresponding
topological markers thus characterize the topology of the interacting phase.
Importantly, the one-particle density matrix is defined in terms of a given
state alone, making the local markers numerically favorable, and providing a
valuable tool for characterizing topology of interacting systems when only the
state itself is available. To demonstrate the practical use of the markers we
use the chiral marker to identify the topology of midspectrum eigenstates of
the Ising-Majorana chain across the transition between the ergodic and
many-body localized phases. We also apply the chiral marker to random states
with a known topology, and compare it with the entanglement spectrum
degeneracy.
