Measurement of Topological Order based on Metric-Curvature Correspondence

TL;DR

This paper establishes a link between topological invariants, curvature, and quantum metric in Dirac models, proposing an experimental method to measure topological order via spectral sum rule violations.

Contribution

It uncovers a metric-curvature correspondence and proposes a novel spectroscopy-based approach to detect topological properties in quantum systems.

Findings

Derived a unified expression for topological invariants across dimensions and classes.

Linked curvature functions to quantum metrics in momentum space.

Suggested an experimental protocol to measure topological order through spectral sum rule violations.

Abstract

A unified expression for topological invariants has been proposed recently to describe the topological order in Dirac models belonging to any dimension and symmetry class. We uncover a correspondence between the curvature function that integrates to this unified topological invariant and the quantum metric that measures the distance between properly defined many-body Bloch states in momentum space. Based on this metric-curvature correspondence, a time-resolved and angle-resolved photoemission spectroscopy experiment is proposed to measure the violation of spectral sum rule caused by a pulse electric field to detect the quantum metric, from which the topological properties of the system may be extracted.