Exploring Grassmann manifolds in topological systems via quantum distance

TL;DR

The paper defines a gauge-invariant quantum distance on Grassmann manifolds $G_{k,n}(\mathbb{C})$ via principal angles and projector-based distances, and uses classical multidimensional scaling to embed the abstract manifold into Euclidean space for visualization. Applying this framework to 2D Chern insulators, 2D topological insulators, a 3D Hopf insulator, and a 3D axion insulator, the authors show that nontrivial topology manifests as $S^2$-like manifolds, holes, and linked preimage contours in the embedded space, with the geometry mirroring topological invariants such as the Chern number, Hopf invariant, and magnetoelectric theta. The approach provides a global, gauge-invariant perspective on quantum geometry, complementary to local Berry curvature analyses, and offers a visualization tool for understanding topological phases across dimensions. This geometric viewpoint could guide intuition and design of materials with robust topological properties by connecting band subspace structure to tangible manifold features in Euclidean embedding.

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.

Figures (15)

• Figure 1: Principal angles $\theta_1$ and $\theta_2$ of two nonparallel planes. The intersection line of two planes determines the first basis $\hat{e}_1$ and $\hat{e}_1^{\prime}$ in two planes, giving $\theta_1=0$. The second principal angles $\theta_2$ is the relative angle between the second basis $\hat{e}_2$ and $\hat{e}_2^{\prime}$. In higher dimensions, there may exist arbitrariness for the basis, so we decide the second basis by making $\theta_2$ as small as possible and use the principle to determine the rest basis.
• Figure 2: Illustrations of quantum disparity of two systems $\Psi$ and $\Phi$ for $4$ filled levels of seven. $\left\{\ket{l} \right\}_{l=1}^7$ is an orthonormal basis. Red colors highlight the differences between $\Psi$ and $\Phi$. The quantum disparity $d_{\Psi,\Phi}$ is independent of the order of the filled/unfilled levels; it is equal to the number of different kets in filled levels between $\Psi$ and $\Phi$.
• Figure 3: Maps of quantum disparity $d_{\vb{k},\vb{k}'}$ for the two-band Chern insulator with $C=1$ in Eq. (\ref{['H2d']}). In each subfigure, the range of $\vb{k}$ is the first BZ and the cyan circle marks the reference point $\vb{k}'$. The red lines highlight the contours of $d=1/2$. The $x$ and $y$ axes are for coordinates $k_x/\pi$ and $k_y/\pi$, respectively.
• Figure 4: Quantum disparity for $C=2$. Same setting as in Fig. \ref{['fig:c1']}.
• Figure 5: Quantum disparity for $C=3$. Same setting as in Fig. \ref{['fig:c1']}.