Topological invariants and corner states for Hamiltonians on a three-dimensional lattice

TL;DR

This paper introduces topological invariants for 3D lattice Hamiltonians with spectral gaps, establishing a link between bulk, edge, and corner states using K-theory, and constructing Hamiltonians from lower-dimensional topological phases.

Contribution

It defines two new topological invariants for 3D lattice Hamiltonians and proves their correspondence, connecting corner states to bulk and edge properties via K-theory.

Findings

Defined two topological invariants for 3D Hamiltonians.

Proved a correspondence between bulk-edge and corner invariants.

Constructed Hamiltonians from 2D and 1D topological insulators.

Abstract

Periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level are considered. By using K-theory applied for the quarter-plane Toeplitz extension, two topological invariants are defined. One is defined for the gapped bulk and edge Hamiltonians, and the non-triviality of the other means that the corner Hamiltonian is gapless. A correspondence between these two invariants is proved. Such gapped Hamiltonians can be constructed from Hamiltonians of 2-D type A and 1-D type AIII topological insulators, and its corner topological invariant is the product of topological invariants of these two phases.