Toeplitz operators on concave corners and topologically protected corner states
Abstract
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.
Keywords
Cite
@article{arxiv.1902.01533,
title = {Toeplitz operators on concave corners and topologically protected corner states},
author = {Shin Hayashi},
journal= {arXiv preprint arXiv:1902.01533},
year = {2019}
}
Comments
v2: references added, typos corrected. 29 pages, 6 figures