We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries. For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by operators of the generic (time-reversal, particle-hole, or chiral) symmetries and additional symmetries, together with gamma matrices in Dirac Hamiltonians representing topological insulators and superconductors. Following Kitaev's approach, we classify gapped phases of non-interacting fermions under additional symmetries by examining all possible distinct Dirac mass terms which can be added to the set of generators of the Clifford algebra. We find that imposing additional symmetries in effect changes symmetry classes and causes shifts in the periodic table of topological insulators and superconductors. Our results are in agreement with the classification under reflection symmetry recently reported by Chiu et al. Several examples are discussed including a topological crystalline insulator with mirror Chern numbers and mirror superconductors.
Periodic Clifford symmetry algebras on flux lattices
实Clifford代数在八个实Altland-Zirnbauer对称类和拓扑相的分类表中起着基础性的作用。在这里，我们提出了另一个优雅的实Clifford代数实现的$d$维无自旋矩形格与$π$通量每个元格。由于$T$ -不变流配置，实Clifford代数被实现为格对称的投射对称代数。值得注意的是，$d$ mod $8$恰好对应于八重Bott周期的实Clifford代数的八个Morita等价类，类似于八个实Altland-Zirnbauer类。Clifford代数的表示理论决定了在一般$k$点和布里渊区的高对称点处的能带结构的简并程度。特别地，我们证明了高对称点处的大简并度为通过各种二聚模式形成新的拓扑态提供了丰富的资源，包括具有双电荷体节环和铰链模的$3$ D高阶半金属态，具有$3$ D表面固体球零模的$4$ D节面半金属态，和$4$ D Möbius拓扑绝缘体具有八重表面节点或四重表面节环。我们的理论可以通过人工晶体的工程$\mathbb{Z}_2$规范场和模拟高维系统的能力在实验上实现。
Selective bulk-boundary correspondence in higher-order topological insulators with anticommuting mirror and chiral symmetries
Indiana University（印第安纳大学）；Department of Physics, Indiana University（印第安纳大学物理系）；Quantum Science and Engineering Center, Indiana University（印第安纳大学量子科学与工程中心）
Topological features of ground states and topological solitons in generalized Su-Schrieffer-Heeger models using generalized time-reversal, particle-hole, and chiral symmetries
