Exploring Grassmann manifolds in topological systems via quantum distance

TL;DR

This paper introduces a method to analyze the geometry of quantum states over parameter spaces by using Grassmann manifolds and quantum distances, revealing topological features in a global, gauge-invariant manner.

Contribution

It presents a novel approach combining quantum distance measures with multidimensional scaling to visualize and analyze topological quantum systems globally.

Findings

Successfully visualized topological features within reconstructed manifolds.

Provided a gauge-invariant framework for quantum state geometry analysis.

Demonstrated the method on various topological system examples.

Abstract

Quantum states defined over a parameter space form a Grassmann manifold. To capture the geometry of the associated gauge structure, gauge-invariant quantities are essential. We employ the projector of a multilevel system to quantify the quantum distance between states. Using the multidimensional scaling method, we transform the quantum distance into a reconstructed manifold embedded in Euclidean space. This approach is demonstrated with examples of topological systems, showcasing their topological features within these manifolds. Our method provides a comprehensive view of the manifold, rather than focusing on local properties.