Topological quantization of ensemble averages

TL;DR

This paper establishes a formal connection between the ensemble average of quantum observables and the index of a Fredholm operator, providing a topological framework for understanding quantized edge currents and conductance.

Contribution

It generalizes a formalism for topological quantization of edge currents to new settings, including atomic wire conductance and edge state detection.

Findings

Differential conductance of atomic wires equals the index of a specific operator.

The formalism links ensemble averages to topological indices under certain conditions.

Potential to identify edge states through topological invariants.

Abstract

We define the current of a quantum observable and, under well defined conditions, we connect its ensemble average to the index of a Fredholm operator. The present work builds on a formalism developed by Kellendonk and Schulz-Baldes \cite{Kellendonk:2004p597} to study the quantization of edge currents for continuous magnetic Schroedinger operators. The generalization given here may be a useful tool to scientists looking for novel manifestations of the topological quantization. As a new application, we show that the differential conductance of atomic wires is given by the index of a certain operator. We also comment on how the formalism can be used to probe the existence of edge states.