Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes

TL;DR

This paper provides a comprehensive theoretical framework for understanding boundary topological insulators and superconductors across all Altland-Zirnbauer symmetry classes, highlighting conditions for nontrivial boundary topology and constructing lattice models for realization.

Contribution

It introduces a unified criterion for boundary topological phases and constructs lattice Hamiltonians for higher-order TIs and TSCs across all symmetry classes.

Findings

Nontrivial boundary topology can emerge at lower-dimensional boundaries with multiple mass terms.

A unified framework extends bulk classification to boundary topologies.

Constructed lattice models enable realization of higher-order TIs and TSCs in various dimensions.

Abstract

In a class of systems, there are gapped boundary-localized states described by a boundary Hamiltonian. The topological classification of gapped boundary Hamiltonians, same as the standard tenfold way for gapped bulk states, can lead to the emergence of boundary topological insulators (TIs) and superconductors (TSCs). In this work, we present a theoretical study of boundary TIs and TSCs of the full Altland-Zirnbauer tenfold symmetry classes. Based on the boundary projection analyses for a $d$-dimensional Dirac continuum model, we demonstrate that nontrivial boundary topology can arise at a $(d-n)$-dimensional boundary if the Dirac model incorporates ($n+1$) mass terms with $0<n<d$ although its bulk and $(d-1)$D, $\cdots$, $(d-n+1)$D boundaries are topologically trivial. Furthermore, we present a unified criterion for the emergence of nontrivial boundary topology by extending bulk classification within the context of the Dirac model, which provides a unified framework for nontrivial bulk and boundary topology. Inspired by the Dirac continuum model analysis, we further construct bulk lattice Hamiltonians for realizing boundary TIs and TSCs of the full Altland-Zirnbauer tenfold symmetry classes, which enables the realization of higher-order TIs and TSCs in arbitrary dimensions with arbitrary orders. We analyze some typical examples of the constructed boundary TIs and TSCs in physical dimensions.