凹角上的Toeplitz算子与拓扑保护角态
Toeplitz operators on concave corners and topologically protected corner states
摘要：
我们认为Toeplitz算子定义在一个凹的角落形状的正方形格的子集。我们得到了这些算子是Fredholm的一个充分必要条件。我们进一步构造了一个指数为1的Fredholm凹角Toeplitz算子.利用这一结果，阐明了四分之一平面和凹角Toeplitz算子的Fredholm指标之间的关系.作为应用，研究了二维AIII类和三维带凹角的A类系统上的体边带隙哈密顿算子的拓扑不变量和角态。明确的例子澄清，这些拓扑不变量依赖于系统的形状。我们讨论了Benalcazar-Bernevig-Hughes的二维哈密顿量，发现只要保持手征对称性，即使打破某些对称性，仍然存在拓扑保护的角态.
英文摘要：
We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar--Bernevig--Hughes' 2-D Hamiltonian and see that there still exists topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.
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