Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
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“真实”布洛赫丛的分类：AI 型拓扑量子系统
DOI：
10.1016/j.geomphys.2014.07.036
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发表时间：
2014
影响因子：
1.5
通讯作者：
Kiyonori Gomi
中科院分区：
文献类型：
--
作者：
Giusppe De Nittis;Kiyonori Gomi
基于“真实的”向量丛的等变同伦性质，对d= 1，2，3，4维的AI型拓扑量子系统进行了分类.这使我们能够产生一个精细的分类，能够照顾到非稳定的制度，这通常是无法通过K理论技术。通过检查对“真实的”线丛进行分类的第二等变上同调群，我们证明了单带无AI或周期量子粒子系统在每个空间维度上不存在非平凡相位。我们还表明，“真实的”线丛的分类足以为AI拓扑量子系统的完全分类在d-3维。在维度d= 4中，不同拓扑相位（对于自由或周期系统）的确定由第二个“真实的”陈类固定，该陈类提供了一个可以用适当映射的度识别的均匀标记。最后，我们为每个给定的拓扑度提供了非平凡的4维自由模型的显式实现。
We provide a classification of type AI topological quantum systems in dimension d= 1, 2, 3, 4 which is based on the equivariant homotopy properties of “Real” vector bundles. This allows us to produce a fine classification able to take care also of the non stable regime which is usually not accessible via K-theoretic techniques. We prove the absence of non-trivial phases for one-band AI free or periodic quantum particle systems in each spatial dimension by inspecting the second equivariant cohomology group which classifies “Real” line bundles. We also show that the classification of “Real” line bundles suffices for the complete classification of AI topological quantum systems in dimension d⩽ 3. In dimension d= 4 the determination of different topological phases (for free or periodic systems) is fixed by the second “Real” Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide explicit realizations of non trivial 4-dimensional free models for each given topological degree.