Mixed-order topology of Benalcazar-Bernevig-Hughes models

TL;DR

This paper explores the topological properties of BBH models, revealing the existence of gapless surface states and topological invariants that depend on boundary conditions, bridging first and higher-order topological insulator behaviors.

Contribution

It identifies new topological invariants and demonstrates the existence of gapless spectra and surface states protected by cubic symmetry in BBH models.

Findings

Existence of gapless spectra of Wilson loops along body diagonals.

Surface states described by massless Dirac fermions.

BBH models can exhibit both first and D-th order topological insulator signatures.

Abstract

Benalcazar-Bernevig-Hughes (BBH) models, defined on $D$-dimensional simple cubic lattice, are paradigmatic toy models for studying $D$-th order topology and corner-localized, mid-gap states. Under periodic boundary conditions, the Wilson loops of non-Abelian Berry connection of BBH models along all high-symmetry axes have been argued to exhibit gapped spectra, which predict gapped surface-states under open boundary conditions. In this work, we identify 1D, 2D, and 3D topological invariants for characterizing higher order topological insulators. Further, we demonstrate the existence of cubic-symmetry-protected, gapless spectra of Wilson loops and surface-states along the body diagonal directions of the Brillouin zone of BBH models. We show the gapless surface-states are described by $2^{D-1}$-component, massless Dirac fermions. Thus, BBH models can exhibit the signatures of first and $D$-th order topological insulators, depending on the details of externally imposed boundary conditions.