Real space topological invariants for metallic systems

A very nice paper was uploaded to arXiv last week: Local invariants identify topological metals. This paper presents a new way to compute the topology and identify robust edge states in gapless and metallic system. Here is my attempt at a summary of their approach:

Background

Introductions to topological phases typically define them in terms of the topological properties of Bloch Hamiltonians. For example, the Chern number is the integral of the Berry curvature over the Brillouin zone. These most familiar definitions assume your system is periodic with Bloch wave eigenstates.

Disorder breaks translation symmetry, such that the eigenstates are no longer Bloch waves. Nevertheless, topological phases remain robust as long as band gaps remain open. To rigorously explain this robustness, topological invariants can be recast in equivalent real space expressions, such as the Chern marker, which can be evaluated even when the system of interest is not periodic. 

The real space topological markers are typically based on the system's Wannier functions or ground state projection operator, which are only short-ranged when the system is gapped. For gapless or metallic systems these functions become long-ranged, leading to non-convergent integrals. But curiously, certain metallic or gapless systems still do exhibit robust edge modes. How can we explain their robustness?

The Localizer

The novel approach taken by Cerjan and Loring in their preprint is based on an operator termed "the localizer" L(x, E). Loosely speaking, the eigenvalues of the localizer measure the strength of perturbation that must be applied to the system's Hamiltonian H in order to shift one of its eigenstates to position x and energy E.

 

In contrast to more conventional measures such as the local density of states, the localizer treats space and energy on the same footing. This allows it to distinguish trivial edge states from topological edge states even in the absence of a bulk energy gap.

For example, the signature of the localizer (difference between the number of its positive and negative eigenvalues) corresponds to the Chern number. The localizer can also be used to detect other kinds of topological states, including corner states of higher order topological insulators.

Outlook

The localizer enables the identification and study of new classes of metallic and gapless topological systems. The authors have suggested that generalizing this approach to higher dimensional topological phases and the full set of symmetry classes is a natural next step. It might also be interesting to consider how one might experimentally measure properties of the localizer. One useful application will be the identification of robust modes in low index contrast photonic systems lacking complete band gaps