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arXiv · 2203.09117

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

Shin Hayashi
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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.

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Shin Hayashi. 2022-03-17. An index theorem for quarter-plane Toeplitz operators via extended symbols and gapped Invariants related to corner states. https://doi.org/10.1007/s00220-022-04600-w

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