An Index Theorem for Quarter-Plane Toeplitz Operators Via Extended Symbols and Gapped Invariants Related to Corner States
COMMUNICATIONS IN MATHEMATICAL PHYSICS(2023)
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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Key words
Topological Insulators,Topological Quantum Computation,Schrödinger Operators
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