Non-local order parameters for states with topological electromagnetic response

TL;DR

This paper introduces non-local order parameters based on electromagnetic responses to identify and characterize Chern insulators and similar topological states, providing a universal diagnostic tool independent of microscopic details.

Contribution

The authors define a class of non-local operators as order parameters for topological states with electromagnetic responses, supported by numerical evaluations and generalizations to higher dimensions.

Findings

Order parameters decay algebraically in topological states

Order parameters decay exponentially in trivial states

Construction can be generalized to 3D topological insulators

Abstract

Chern insulators are states of matter characterized by a quantized Hall conductance, gapless edge modes but also a singular response to monopole configurations of an external electromagnetic field. In this paper, we describe the nature of such a singular response and show how it can be used to define a class of operators acting as non-local order parameters. These operators characterize the Chern-insulator states in the following way: for a given state, there exists a corresponding operator which has an algebraically decaying two-point function in that particular state, while it decays exponentially in all other states. The behaviour of the order parameter is defined only in terms of the electromagnetic response, and not from any microscopic properties, and we therefore claim to have found a generic order parameter for the Chern insulating states. We support this claim by numerically evaluating the order parameters for different insulating states. We also show how our construction can be generalized to other states with topological electromagnetic response, and use the states with a quantized magnetoelectric effect in three dimensions as an example. Besides providing novel insights into topological states of matter, our construction can be exploited to efficiently diagnose such states numerically.