Symmetry-protected topological phases in noninteracting fermion systems
Symmetry-protected topological phases in noninteracting fermion systems
复制标题
非相互作用费米子系统中对称保护的拓扑相
DOI：
10.1103/physrevb.85.085103
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发表时间：
2012-02-09
影响因子：
3.7
通讯作者：
Wen, Xiao-Gang
中科院分区：
文献类型：
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作者：
Wen, Xiao-Gang
对称性保护拓扑相是具有一定对称性的带隙量子相，如果对称性被破坏，这些带隙量子相可以光滑地连接到同一个平凡乘积态。对于具有时间反演（T）、电荷共轭（C）和/或U（1）（N）对称性的非相互作用费米子系统，全对称群可以取决于这些对称性操作之间的关系，例如T N T^{-1}= N或T N T^{-1}= -N。结果表明，不同对称群的费米子系统的SPT相有不同的分类。在本文中，我们使用Kitaev的K-理论的方法来分类这些可能的对称群的带隙自由费米子相位。特别地，我们可以将U（1）视为自旋旋转。我们发现，在偶数维中，具有S_z自旋旋转对称性的超导体可以用Z分类，而在奇数维中，具有时间反转和S_z自旋旋转对称性的超导体可以用Z分类。我们表明，所有10类带隙自由费米子相位可以实现的电子系统具有一定的对称性。我们还指出，为了正确地描述费米子系统的对称性，我们需要指定它的完整对称群，包括费米子数宇称变换（-）^N。全对称群实际上是一个射影对称群。
Symmetry protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For non-interacting fermion systems with time reversal (T), charge conjugation (C), and/or U(1) (N) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as T N T^{-1}= N or T N T^{-1}= -N. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free fermion phases for those possible symmetry groups. In particular, we can view the U(1) as a spin rotation. We find that superconductors with the S_z spin rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the S_z spin rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free fermion phases can be realized by electron systems with certain symmetries. We also point out that to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (-)^N. The full symmetry group is actually a projective symmetry group.