Classification of topological invariants related to corner states

Abstract

We discuss some bulk-surface gapped Hamiltonians on a lattice with corners and propose a periodic table for topological invariants related to corner states aimed at studies of higher-order topological insulators. Our table is based on four things: (1) the definition of topological invariants, (2) a proof of their relation with corner states, (3) computations of K-groups and (4) a construction of explicit examples.

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

“…As in [20,22], from the pair of invertible half-space operators H 0 and H ∞ , we can define an element of the K-group K i (S 0,∞ ⊗ C(T n−2 )) or KO i (S 0,∞ ⊗ C(T n−2 ), τ S ⊗ τ c n−2 ), and I ♠ Gapless (H) is the image of this element through the boundary map ∂ qT of the long exact sequence of K-theory for C * -algebras or KOtheory for C * ,τ -algebras associated with the extension (7) or (24). Therefore, by Theorem 4.1 and Theorem 5.4, we obtain the following result.…”

“…For this purpose, we use Atiyah's KR-theory for spaces equipped with involutions [3] and Boersema-Loring's formulation for the KO-theory of real C * -algebras [10]. Index theory for quarter-plane Toeplitz operators preserving some real structures and application to topological corner states are discussed in [22], which we mainly follow. 5.1.…”

“…, where i and H E are as indicated in Table 2. Theorem 6.2 provides a geometric formulation for a relation between a gapped topological invariant and corner states in [20,22], though, since our three sphere S3 is not smooth as the boundary of D 2 × D 2 , an integration formula for numerical gapped invariants, like integration of the Berry curvature for the first Chern number, is still missing. At this stage, we simply note the following understanding of numerical gapped invariants for two-dimensional class AIII systems with a corner and three-dimensional class A systems with a hinge.…”

“…In this section, we discuss some bulk-edge gapped Hamiltonians on a lattice with a codimension-two corner or hinge. By using Theorem 4.1 and Theorem 5.4, we provide more geometric way to formulate a relation between topological invariants for such gapped Hamiltonians and corner/hinge states than that in [20,22].…”

“…For matrix factorizations of rational matrix functions on the unit circle in the complex plane, an algorithm is known [18,12,17] and the finite hopping range condition is assumed correspondingly. In [22], a classification of topological invariants related to corner states in each of the Altland-Zirnbauer classes are proposed based on index theory, where Boersema-Loring's formulation of KO-theory for real C * -algebras [10] is employed. Since topological corner states are one motivation of this work, some parts of the discussions in this paper are organized in this framework.…”