Higher Order Bosonic Topological Phases in Spin Models

TL;DR

This paper introduces bosonic models exhibiting higher order topological phases with protected corner modes, extending the concept beyond fermionic systems and demonstrating their relation to Majorana fermions.

Contribution

The paper presents two spin models for bosonic second-order topological phases protected by z_2 d7 z_2 symmetry, linking them to Majorana fermions.

Findings

Models host gapped, topological edges and protected corner modes.

Models are related to bilayer Majorana fermions forming a fermionic topological phase.

Extension to three dimensions suggests possible third-order topological phases.

Abstract

We discuss an extension of higher order topological phases to include bosonic systems. We present two spin models for a second-order topological phase protected by a global $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional $\mathbb{Z}_2\times\mathbb{Z}_2$ topological phase, while the other is built from more conventional spin-couplings (XY or Heisenberg). These models host gapped, but topological, edges, and protected corner modes that fall into a projective representation of the symmetry. Using Jordan-Wigner transformations we show that our models are both related to a bilayer of free Majorana fermions that form a fermionic second-order topological phase. We also discuss how our models can be extended to three-dimensions to form a third-order topological phase.