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

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.

32 pages, 4 figures

References in corpus (3)

• Chiral-Symmetric Higher-Order Topological Phases of Matter
• Toeplitz operators on concave corners and topologically protected corner states
• Classification of topological invariants related to corner states

Cited by in corpus (2)

• Interfaces of discrete systems - spectral and index properties
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values