A Quantized Inter-level Character in Quantum Systems

TL;DR

This paper introduces a new quantized inter-band character in quantum systems, analogous to the Euler characteristic, which is gauge invariant and relates to Berry curvature differences, extending to degenerate systems.

Contribution

It defines a novel quantized inter-band character in quantum systems, providing a gauge-invariant topological measure analogous to the Euler characteristic, and generalizes it to degenerate systems.

Findings

The inter-band character $ heta$ is gauge invariant and quantized as an integer.

It is analogous to the Euler characteristic via the Gauss-Bonnet theorem.

The concept extends to quantum degenerate systems.

Abstract

For a quantum system subject to external parameters, the Berry phase is an intra-level property, which is gauge invariant module $2\pi$ for a closed loop in the parameter space and generally is non-quantized. In contrast, we define a inter-band character $\Theta$ for a closed loop, which is gauge invariant and quantized as integer values. It is a quantum mechanical analogy of the Euler character based on the Gauss-Bonnet theorem for a manifold with a boundary. The role of the Gaussian curvature is mimicked by the difference between the Berry curvatures of the two levels, and the counterpart of the geodesic curvature is the quantum geometric potential which was proposed to improve the quantum adiabatic condition. This quantized inter-band character is also generalized to quantum degenerate systems.