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Topological properties of physical systems can lead to robust behaviors that are insensitive to microscopic details.
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Note that evolution operators are only defined up to the choice of the basis and two unitary matrices U k U_{k} and U k ′ = V k − 1 U k V k U^{\prime}_{k}=V^{-1}_{k}U_{k}V_{k} ( V k V_{k} is a unitary matrix) that are related by k k dependent similarity transformations represent the same evolution. It is straightforward to show that ν 1 \nu_{1} is invariant under such similarity transformations, and therefore, is independent of the choice of basis
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When there are degeneracies in the spectrum, one should choose the definitions of { ε k , α } \{\varepsilon_{k,\alpha}\} such that they become smooth as a function of quasi-momentum k k
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