A Gauss-Bonnet Theorem for Quantum States: Gauss Curvature and Topology in the Projective Hilbert Space

TL;DR

This work develops a gauge-invariant differential-geometric framework for quantum states by formulating the quantum metric and curvature from eigenprojectors, enabling a direct analysis of the quantum-state manifold in Bloch bands. It reveals a closed singular curve where the metric degenerates and introduces a front-based approach with a signed area form to handle these folds, culminating in a generalized Gauss–Bonnet relation that ties the total signed Gauss curvature to the Chern number. In a two-band model, the Gauss curvature is constant on regular regions (K_G = 2) while folds mediate the relation between quantum volume and topology, with the Berry curvature appearing as the signed area density (Ω = $\bar{\lambda}$) and $\det g = Ω^2$. The framework unifies quantum metric and Berry curvature within a geometric, topological viewpoint and suggests pathways to extend to multi-band and non-Hermitian systems, with potential links to measurable higher-order responses.

Abstract

Geometry and topology are fundamental to modern condensed matter physics, but their precise connection in quantum systems remains incompletely understood. Here, we develop an analytical scheme for calculating the curvature of the quantum metric of Bloch bands. Using a gauge-invariant formulation based on eigenprojectors, we construct the full Riemannian geometry of the quantum-state manifold and apply it to a two-dimensional two-band model. We find that the Gauss curvature is constant over regular regions, but the manifold inevitably develops a closed curve of singular points where the metric tensor degenerates. These singularities obstruct the conventional Gauss-Bonnet theorem. By introducing the notion of a front and a signed area form, we derive a generalized Gauss-Bonnet relation that includes a singular curvature term defined along the fold curve. This result establishes a direct, quantized link between the total signed Gauss curvature and the Chern number, providing a unified geometric interpretation of Berry curvature and quantum metric. This framework bridges differential geometry and topological band theory, revealing how singular folds mediate the discrepancy between quantum volume and topological charge.

Figures (5)

• Figure 1: The determinant of the metric tensor $\det g$ for the two-band model. The red lines are singular points where $\det g=0$.
• Figure 2: A transformation from a torus to a sphere. In the transparent tori, the dark areas are the front face made of three projections. In the most right object, the Whitney cusps are marked by red circles.
• Figure 3: (a) The singular curve (blue lines) and the kernel vectors (in red) at singular points. The dashed lines circumscribe the first Brillouin zone. (b) The manifold of the quantum system (the sphere) and the singular-value curve (the blue line). Four cusp points are visible as the image of $(k_x,k_y)=\pm(\pi/2,\pm\pi/2)$. (c) and (d) are the $\vb{k}$ points and their maps, respectively. The domain is separated into $M_+$ and $M_-$. In addition, $M_+$ is the union of three colorful regions (1, 2, and 3). A trajectory in $k$-space (red lines) in the direction as the gray arrow shows is in (c) and its image on the sphere (red dots) is in (d). When the curve passes the singular points (green triangles), the image curve is deflected at $\widetilde{\Sigma}$. Each point in $\widetilde{M}_3$ has three preimages (illustrated by blue $\times$ and red $+$ markers), while every point in $\widetilde{M}_1$ has one preimage (by black $\ast$).
• Figure 4: The singular curvature $\kappa_s$ and the length element $ds$ in the line integral along the singular curve. The angle $\theta_k$ is measured relative to the $k_x$-axis around $(\pi,\pi)$. While the singular curvature diverges at the peaks, the length element simultaneously approaches zero.
• Figure 5: (a) The adapted coordinates $(u,v)$ along the singular curve. (b) The normalized tangent vectors $(\tilde{u}^1,\tilde{u}^2)$ from Eq. (\ref{['u_cho']}), and (c) the normalized kernel vector $(\tilde{v}^1,\tilde{v}^2)$ from Eq. (\ref{['v_cho']}) in the singular curve. The angle $\theta_k$ is relative to the $k_x$-axis around $(\pi,\pi)$. Our choice has continuous $u$'s but has $u$'s discontinuous at the preimages of the cusps. This choice can produce a consistent positive frame: $\mathrm{sgn}(u\wedge v)>0$.