TY - JOUR

T1 - An Index Theorem for Quarter-Plane Toeplitz Operators via Extended Symbols and Gapped Invariants Related to Corner States

AU - Hayashi, Shin

N1 - Funding Information:
This work was supported by JSPS KAKENHI (Grant Nos. JP17H06461, JP19K14545) and JST PRESTO (Grant No. JPMJPR19L7).
Publisher Copyright:
© 2022, The Author(s).

PY - 2022

Y1 - 2022

N2 - 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–Kreĭn 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.

AB - 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–Kreĭn 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.

UR - http://www.scopus.com/inward/record.url?scp=85144175798&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85144175798&partnerID=8YFLogxK

U2 - 10.1007/s00220-022-04600-w

DO - 10.1007/s00220-022-04600-w

M3 - Article

AN - SCOPUS:85144175798

SN - 0010-3616

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

ER -