Quantized interlevel character in quantum systems

@article{Xu2018QuantizedIC,
  title={Quantized interlevel character in quantum systems},
  author={Chao Xu and Jianda Wu and Congjun Wu},
  journal={Physical Review A},
  year={2018},
  volume={97},
  pages={032124}
}
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… 

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