Method of computing Hall conductivities on discrete Brillouin zone-topological quantum numbers on lattices-

本研究的目的是：（1）发展一种有效的计算拓扑不变量的方法;（2）将其应用于实际材料。2006年，我们在计算量子霍尔效应的第一个陈数方面取得了重大进展。2007年，我们成功地计算了表征量子自旋霍尔效应的Z2拓扑不变量。主要工作如下：1）石墨烯的量子霍尔效应：计算了石墨烯在整个能区的霍尔电导。我们证明了在van-Hove奇异性之间存在Dirac类准粒子。2）计算量子自旋霍尔效应的Z2不变量的基本算法：我们发展了一种有效的算法，并将其应用于一个简单但典型的模型，即，具有自旋轨道耦合的石墨烯。我们已经表明，我们的方法确实足够简单，可以对石墨烯的两相进行分类。3）上述算法在三维材料中的应用：我们计算了体Bi和Sb以及它们的薄膜的Z2不变量。由于这些材料是半金属，我们不期望量子自旋霍尔效应。但我们认为它们的薄膜是绝缘体，并可能表现出量子自旋霍尔效应。

The purposes of this research are 1) to develop an efficient method of computing topological invariants and 2) its application to realistic materials. In 2006, we had a big progress in computing the first Chern numbers for quantum Hall effect. In 2007, we succeeded in computing the Z2 topological invariant which characterizes the quantum spin Hall effect. What we have achieved are summarized as follows:1) Quantum Hall effect in graphene: We have computed the Hall conductance of graphene over the whole energy region. We have shown that in between the van-Hove singuralities, Dirac -like quasi-particles exist.2) Basic algorism of computing Z2 invariant for quantum spin Hall effect: We have developed an efficient algorism and applied it to a simple but typical model, I.e., graphene with spin-orbit couplings. We have shown that out method is indeed simple enough to classify the two phases of graphene.3) Application of the above algorism to three dimensional materials: We have computed the Z2 invariants for the bulk Bi and Sb as well as their thin films. Since these materials are semi-metal, we do not expect the quantum spin Hall effect. But we have suggested that their thin films become insulators and could show the quantum spin Hall effect.