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Abstract

For a quantum system subject to external parameters, the Berry phase is an intralevel property, which is gauge invariant module 2 pi for a closed loop in the parameter space and generally is nonquantized. In contrast, we define an interband 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 interband character is also generalized to quantum degenerate systems.