Wannier函数在Chern和量子Hall绝缘体中的最优衰减
Optimal decay of Wannier functions in Chern and Quantum Hall insulators
摘要：
我们研究了独立电子在周期性背景中的局域化性质，可能包括周期性磁场，例如在陈氏绝缘体和量子霍尔系统中。一般来说，由于哈密顿量的谱是绝对连续的，因此局域化的特征是与费米能以下的布洛赫带相关的复合（磁性）瓦尼尔函数的衰减，如$|x| \rightarrow \infty$，这被认为是在谱隙中。我们证明了本地化二分法的有效性，在以下意义上：要么存在指数本地化的复合Wannier函数，相应的系统是在一个平凡的拓扑相位与消失的霍尔电导率，或任何复合Wannier函数的衰减是这样的平方位置算子的期望值，或等效的Marzari-Vanderbilt本地化功能，是$+ \infty$。在后一种情况下，布洛赫丛在拓扑上是非平凡的，并且人们期望非零的霍尔电导率。
英文摘要：
We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e.g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as $|x| \rightarrow \infty$, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is $+ \infty$. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.
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