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

TL;DR

This paper introduces a method using the one-particle density matrix to characterize the topology of interacting and disordered quantum states, enabling analysis without explicit Hamiltonian knowledge.

Contribution

It extends topological markers to interacting systems via the density matrix, allowing topology characterization solely from the state, even in disordered and many-body localized phases.

Findings

Successfully identified topology in the Ising-Majorana chain across phase transitions.

Validated the marker against known topological states and entanglement spectrum degeneracies.

Demonstrated numerical efficiency and applicability to disordered and interacting systems.

Abstract

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.