The geometric and topological interpretation of Berry phase on a torus

TL;DR

This paper provides a clear geometric and topological interpretation of Berry phase on a torus, linking curvature to Gaussian curvature and illustrating the phase as a rotation angle.

Contribution

It offers a concrete system to visualize Berry phase, connecting geometric properties with topological invariants using a torus parameter space.

Findings

Curvature magnitude equals Gaussian curvature.

Berry phase corresponds to the angle between initial and final vectors.

Gauge transformation is a rotation of the basis.

Abstract

Illustration of the geometric and topological properties of Berry phase is often in an obscure and abstract language of fiber bundles. In this article, we demonstrate these properties with a lucid and concrete system whose parameter space is a torus. The instantaneous eigenstate is regarded as a basis. And the corresponding connection and curvature are calculated respectively. Furthermore, we find the magnitude of curvature is exactly the Gaussian curvature, which shows its local property. The topological property is reflected by the integral over the torus due to Gauss-Bonnet theorem. When we study the property of parallel transportation of a vector over a loop, we make a conclusion that the Berry phase is just the angle between the final and initial vectors. And we also illuminate the geometric meaning of gauge transformation, which is just a rotation of basis.