论文标题
拓扑与合成维度中的定位
Topology vs localization in synthetic dimensions
论文作者
论文摘要
由合成维度的量子模拟最近发展的动机，例如在Ultracold Atoms的光学晶格中，我们在这里讨论了$ D $ d $ d $ 4 $ 4 $的量子系统，重点关注被占用能量状态的拓扑。我们通过询问是否可以通过平滑和周期性的BLOCH函数来跨越差距的光谱子空间来执行此分析，这对应于位置空间中的局部Wannier函数。通过在维度中构造这些BLOCH的功能，我们表明，如果它们必须是正统的，那么通常，它们的存在被基础Bloch束的前两个Chern类所阻碍，而第二个Chern类则在4维情况下进行表征。如果正常限制放松，我们将展示如何通过最多包含$ m+2 $ bloch功能的parseval框架来跨越$ m $占用的能量频段。
Motivated by recent developments in quantum simulation of synthetic dimensions, e.g. in optical lattices of ultracold atoms, we discuss here $d$-dimensional periodic, gapped quantum systems for $d \le 4$, with focus on the topology of the occupied energy states. We perform this analysis by asking whether the spectral subspace below the gap can be spanned by smooth and periodic Bloch functions, corresponding to localized Wannier functions in position space. By constructing these Bloch functions inductively in the dimension, we show that if they are required to be orthonormal then in general their existence is obstructed by the first two Chern classes of the underlying Bloch bundle, with the second Chern class characterizing in particular the 4-dimensional situation. If the orthonormality constraint is relaxed, we show how $m$ occupied energy bands can be spanned by a Parseval frame comprising at most $m+2$ Bloch functions.
