Local Topological Markers for Disordered, Interacting, and Mixed States

The topology of crystalline insulators and superconductors is characterized by established momentum-space invariants such as the Chern number. In amorphous and other disordered systems, however, momentum is no longer a good quantum number, and recent observations of topological edge states in amorphous insulators highlight the need for real-space tools that do not rely on periodicity. In even spatial dimensions, the Chern marker provides a real-space version of the Chern number, obtained by Fourier transforming the Chern character. In odd dimensions, the relevant invariants—the winding number and the Z2 invariant—are defined through gauge-dependent differential forms that cannot be Fourier transformed into local real-space expressions. This has left odd-dimensional real-space topology without an analogue of the Chern marker.

 

In this talk, I will explain why the differential forms that define odd-dimensional topological invariants cannot be Fourier transformed into real space, and how we resolve this and develop local topological markers in odd spatial dimensions. Our approach is based on dimensional reduction: we express the even-dimensional Chern character in terms of a one-parameter family of projectors P(theta) interpolating between a trivial state and the state of interest. Treating theta as an auxiliary coordinate and integrating over it yields closed-form, real-space expressions for odd-dimensional topological markers—the chiral marker (a local Z invariant that recovers the winding number) and the Chern–Simons marker (a local Z2 invariant capturing non-chiral phases). Together, these markers characterize four of the five Altland–Zirnbauer symmetry classes that can host topological phases in each odd spatial dimension.

 

An advantage of our formalism is that it characterizes the topology of the state itself, independent of the parent Hamiltonian. It also extends to interacting systems, provided the one-particle density matrix has a spectral gap that allows it to be adiabatically flattened to a projector. I will show how these markers can be used to identify the topology of amorphous three-dimensional topological insulators and superconductors, as well as mid-spectrum many-body localized states in the interacting Ising–Majorana model. Finally, I will explain how the same framework applies to mixed Gaussian states with a purity gap, enabling real-space diagnostics of topology at finite temperature or in open systems. Together, these results provide a general, practical, and state-based approach to identifying topological phases in disordered, interacting, and non-equilibrium quantum matter.

 

References:

Phys. Rev. Lett. 129, 277601 (2022)

Phys. Rev. Research 6, L032045 (2024)

EPL 142, 16001 (2023)

arXiv:2511.xxxx (in preparation)