An index theorem for quarter-plane Toeplitz operators via extended symbols and gapped Invariants related to corner states

TL;DR

This paper develops an index theorem for quarter-plane Toeplitz operators using extended symbols, linking topological invariants to corner states in lattice Hamiltonians.

Contribution

It introduces a novel index formula for quarter-plane Toeplitz operators via symbol extension and relates these to topological invariants of corner states in lattice models.

Findings

Derived a formula expressing Fredholm indices through extended symbols.

Extended symbols from 2D torus to 3D sphere for index calculation.

Linked topological invariants to corner states in bulk-edge Hamiltonians.

Abstract

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg-Krein theory for matrix factorizations, we extend the symbols defined originally on a two-dimensional torus to some three-dimensional sphere and derive a formula to express their Fredholm indices through extended symbols. Variants for families of (self-adjoint) Fredholm quarter-plane Toeplitz operators and those preserving real structures are also included. For some bulk-edge gapped single-particle Hamiltonians of finite hopping range on a discrete lattice with a codimension-two right angle corner, topological invariants related to corner states are provided through extensions of bulk Hamiltonians.