# Toeplitz operators on concave corners and topologically protected corner states

## TL;DR

This paper studies Toeplitz operators on concave corners, establishing conditions for their Fredholm property, and explores topologically protected corner states in certain 2D and 3D Hamiltonian systems, revealing shape-dependent invariants.

## Contribution

It provides a necessary and sufficient condition for Fredholmness of concave corner Toeplitz operators and links their indices to topological invariants in physical systems.

## Key findings

- Fredholm condition characterized for concave corner Toeplitz operators
- Constructed a Fredholm operator with index one for concave corners
- Identified topologically protected corner states in specific Hamiltonians despite symmetry breaking

## 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.